Given's Rotation SVD example, simplified

Percentage Accurate: 75.9% → 99.7%

Time: 10.9s

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: 75.9% 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: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 2.0)
     (fma 0.125 (* x x) (* -0.0859375 (pow x 4.0)))
     (/ 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 = fma(0.125, (x * x), (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = 1.0 / ((1.0 + 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 = fma(0.125, Float64(x * x), Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 + sqrt(Float64(0.5 + t_0))) / Float64(0.5 - t_0)));
	end
	return tmp
end

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[(0.125 * N[(x * x), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $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 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 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%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 2.0)
     (fma 0.125 (* x x) (* -0.0859375 (pow x 4.0)))
     (/ (- 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 = fma(0.125, (x * x), (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + 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 = fma(0.125, Float64(x * x), Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return 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[(0.125 * N[(x * x), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $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 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 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%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (fma 0.125 (* x x) (* -0.0859375 (pow x 4.0)))
   (/ 1.0 (/ 1.0 (- 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 = fma(0.125, (x * x), (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = 1.0 / (1.0 / (1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))))));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 98.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}} \]

      2. expm1-udef 98.5%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} - 1}} \]

      3. clear-num 98.5%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}}\right)} - 1} \]

      4. metadata-eval 98.5%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{\left(1 - 0.5\right)} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)} - 1} \]

      5. associate--r+ 98.5%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{1 - \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)} - 1} \]

      6. metadata-eval 98.5%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\frac{1}{\frac{\color{blue}{1 \cdot 1} - \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)} - 1} \]

      7. add-sqr-sqrt 98.4%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\frac{1}{\frac{1 \cdot 1 - \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)} - 1} \]

      8. flip-- 98.5%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\right)} - 1} \]

   9. Applied egg-rr 98.5%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(\frac{1}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)} - 1}} \]
   10. Step-by-step derivation

       1. expm1-def 98.5%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)\right)}} \]

       2. expm1-log1p 98.5%

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

   11. Simplified 98.5%

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

4. Final simplification 99.2%

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

Alternative 4: 98.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* 0.125 (* 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 = 0.125 * (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 = 0.125 * (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 = 0.125 * (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(0.125 * Float64(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 = 0.125 * (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[(0.125 * N[(x * x), $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 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.1%

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

Alternative 5: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (fma 0.125 (* x x) (* -0.0859375 (pow x 4.0)))
   (- 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 = fma(0.125, (x * x), (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.2%

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

Alternative 6: 99.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (/ (- 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ -0.5 x)))))
   (if (<= x 1.2)
     (* 0.125 (* x x))
     (* (/ 1.0 (- -1.0 (sqrt (+ 0.5 (/ 0.5 x))))) (+ -0.5 (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = (1.0 / (-1.0 - sqrt((0.5 + (0.5 / x))))) * (-0.5 + (0.5 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.2d0)) then
        tmp = (0.5d0 - ((-0.5d0) / x)) / (1.0d0 + sqrt((0.5d0 + ((-0.5d0) / x))))
    else if (x <= 1.2d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = (1.0d0 / ((-1.0d0) - sqrt((0.5d0 + (0.5d0 / x))))) * ((-0.5d0) + (0.5d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (-0.5 / x))));
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = (1.0 / (-1.0 - Math.sqrt((0.5 + (0.5 / x))))) * (-0.5 + (0.5 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.2:
		tmp = (0.5 - (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (-0.5 / x))))
	elif x <= 1.2:
		tmp = 0.125 * (x * x)
	else:
		tmp = (1.0 / (-1.0 - math.sqrt((0.5 + (0.5 / x))))) * (-0.5 + (0.5 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(0.5 - Float64(-0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(-0.5 / x)))));
	elseif (x <= 1.2)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / Float64(-1.0 - sqrt(Float64(0.5 + Float64(0.5 / x))))) * Float64(-0.5 + Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.2)
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	elseif (x <= 1.2)
		tmp = 0.125 * (x * x);
	else
		tmp = (1.0 / (-1.0 - sqrt((0.5 + (0.5 / x))))) * (-0.5 + (0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(0.5 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(-1.0 - N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996

   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 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

      1. flip-- 97.4%

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

      2. div-inv 97.4%

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

      3. metadata-eval 97.4%

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

      4. add-sqr-sqrt 98.8%

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

      5. sub-neg 98.8%

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

      6. associate--r+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-neg-frac 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-neg-frac 98.8%

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

      12. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   10. Simplified 98.9%

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

   if -1.19999999999999996 < x < 1.19999999999999996

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]

   if 1.19999999999999996 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around -inf 94.8%

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

      1. associate-*r/ 94.8%

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

      2. metadata-eval 94.8%

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

   6. Simplified 94.8%

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

      1. flip-- 94.8%

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

      2. div-inv 94.8%

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

      3. metadata-eval 94.8%

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

      4. add-sqr-sqrt 96.3%

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

      5. sub-neg 96.3%

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

      6. associate--r+ 96.3%

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

      7. metadata-eval 96.3%

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

      8. distribute-neg-frac 96.3%

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

      9. metadata-eval 96.3%

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

      10. sub-neg 96.3%

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

      11. distribute-neg-frac 96.3%

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

      12. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. associate-*r/ 96.3%

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

      2. *-rgt-identity 96.3%

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

   10. Simplified 96.3%

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

   12. Applied egg-rr 98.1%

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

       1. associate-*r/ 98.1%

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

       2. *-rgt-identity 98.1%

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

       3. distribute-neg-in 98.1%

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

       4. metadata-eval 98.1%

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

       5. unsub-neg 98.1%

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

   14. Simplified 98.1%

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

       1. clear-num 98.1%

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

       2. associate-/r/ 98.1%

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

       3. +-commutative 98.1%

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

   16. Applied egg-rr 98.1%

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

4. Final simplification 99.1%

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

Alternative 7: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + \frac{0.5}{x}}{-1 - \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (if (<= x 1.2)
     (* 0.125 (* x x))
     (/ (+ -0.5 (/ 0.5 x)) (- -1.0 (sqrt (+ 0.5 (/ 0.5 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = (-0.5 + (0.5 / x)) / (-1.0 - sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    else if (x <= 1.2d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = ((-0.5d0) + (0.5d0 / x)) / ((-1.0d0) - sqrt((0.5d0 + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	} else if (x <= 1.2) {
		tmp = 0.125 * (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 x <= -1.55:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	elif x <= 1.2:
		tmp = 0.125 * (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 (x <= -1.55)
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	elseif (x <= 1.2)
		tmp = Float64(0.125 * Float64(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 (x <= -1.55)
		tmp = 0.5 / (1.0 + sqrt(0.5));
	elseif (x <= 1.2)
		tmp = 0.125 * (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[x, -1.55], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(0.125 * N[(x * x), $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 3 regimes

2. if x < -1.55000000000000004

   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-- 98.4%

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

      2. div-inv 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 97.6%

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

   if -1.55000000000000004 < x < 1.19999999999999996

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]

   if 1.19999999999999996 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around -inf 94.8%

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

      1. associate-*r/ 94.8%

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

      2. metadata-eval 94.8%

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

   6. Simplified 94.8%

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

      1. flip-- 94.8%

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

      2. div-inv 94.8%

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

      3. metadata-eval 94.8%

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

      4. add-sqr-sqrt 96.3%

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

      5. sub-neg 96.3%

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

      6. associate--r+ 96.3%

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

      7. metadata-eval 96.3%

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

      8. distribute-neg-frac 96.3%

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

      9. metadata-eval 96.3%

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

      10. sub-neg 96.3%

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

      11. distribute-neg-frac 96.3%

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

      12. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. associate-*r/ 96.3%

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

      2. *-rgt-identity 96.3%

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

   10. Simplified 96.3%

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

   12. Applied egg-rr 98.1%

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

       1. associate-*r/ 98.1%

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

       2. *-rgt-identity 98.1%

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

       3. distribute-neg-in 98.1%

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

       4. metadata-eval 98.1%

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

       5. unsub-neg 98.1%

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

   14. Simplified 98.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + \frac{0.5}{x}}{-1 - \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]

Alternative 8: 99.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (/ (- 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ -0.5 x)))))
   (if (<= x 1.2)
     (* 0.125 (* x x))
     (/ (+ -0.5 (/ 0.5 x)) (- -1.0 (sqrt (+ 0.5 (/ 0.5 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = (-0.5 + (0.5 / x)) / (-1.0 - sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.2d0)) then
        tmp = (0.5d0 - ((-0.5d0) / x)) / (1.0d0 + sqrt((0.5d0 + ((-0.5d0) / x))))
    else if (x <= 1.2d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = ((-0.5d0) + (0.5d0 / x)) / ((-1.0d0) - sqrt((0.5d0 + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (-0.5 / x))));
	} else if (x <= 1.2) {
		tmp = 0.125 * (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 x <= -1.2:
		tmp = (0.5 - (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (-0.5 / x))))
	elif x <= 1.2:
		tmp = 0.125 * (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 (x <= -1.2)
		tmp = Float64(Float64(0.5 - Float64(-0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(-0.5 / x)))));
	elseif (x <= 1.2)
		tmp = Float64(0.125 * Float64(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 (x <= -1.2)
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	elseif (x <= 1.2)
		tmp = 0.125 * (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[x, -1.2], N[(N[(0.5 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(0.125 * N[(x * x), $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 3 regimes

2. if x < -1.19999999999999996

   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 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   6. Simplified 97.4%

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

      1. flip-- 97.4%

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

      2. div-inv 97.4%

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

      3. metadata-eval 97.4%

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

      4. add-sqr-sqrt 98.8%

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

      5. sub-neg 98.8%

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

      6. associate--r+ 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-neg-frac 98.8%

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

      9. metadata-eval 98.8%

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

      10. sub-neg 98.8%

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

      11. distribute-neg-frac 98.8%

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

      12. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   10. Simplified 98.9%

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

   if -1.19999999999999996 < x < 1.19999999999999996

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]

   if 1.19999999999999996 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around -inf 94.8%

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

      1. associate-*r/ 94.8%

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

      2. metadata-eval 94.8%

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

   6. Simplified 94.8%

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

      1. flip-- 94.8%

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

      2. div-inv 94.8%

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

      3. metadata-eval 94.8%

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

      4. add-sqr-sqrt 96.3%

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

      5. sub-neg 96.3%

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

      6. associate--r+ 96.3%

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

      7. metadata-eval 96.3%

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

      8. distribute-neg-frac 96.3%

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

      9. metadata-eval 96.3%

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

      10. sub-neg 96.3%

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

      11. distribute-neg-frac 96.3%

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

      12. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. associate-*r/ 96.3%

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

      2. *-rgt-identity 96.3%

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

   10. Simplified 96.3%

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

   12. Applied egg-rr 98.1%

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

       1. associate-*r/ 98.1%

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

       2. *-rgt-identity 98.1%

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

       3. distribute-neg-in 98.1%

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

       4. metadata-eval 98.1%

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

       5. unsub-neg 98.1%

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

   14. Simplified 98.1%

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

4. Final simplification 99.1%

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

Alternative 9: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (if (<= x 1.2) (* 0.125 (* x x)) (- 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.55d0)) then
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    else if (x <= 1.2d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 1.0d0 - sqrt((0.5d0 + (0.5d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.55:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	elif x <= 1.2:
		tmp = 0.125 * (x * x)
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	elseif (x <= 1.2)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = 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 (x <= -1.55)
		tmp = 0.5 / (1.0 + sqrt(0.5));
	elseif (x <= 1.2)
		tmp = 0.125 * (x * x);
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   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-- 98.4%

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

      2. div-inv 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 97.6%

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

   if -1.55000000000000004 < x < 1.19999999999999996

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]

   if 1.19999999999999996 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around inf 96.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \]

Alternative 10: 98.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 1.55)))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (* 0.125 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.55)) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 1.55d0))) then
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    else
        tmp = 0.125d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.55)) {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 1.55):
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	else:
		tmp = 0.125 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 1.55))
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(0.125 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 1.55)))
		tmp = 0.5 / (1.0 + sqrt(0.5));
	else
		tmp = 0.125 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1.55000000000000004 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 97.0%

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

   if -1.55000000000000004 < x < 1.55000000000000004

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternative 11: 97.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 1.55))) (- 1.0 (sqrt 0.5)) (* 0.125 (* x x))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 1.55d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 0.125d0 * (x * x)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 1.55):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 0.125 * (x * x)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1.55000000000000004 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around inf 95.6%

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

   if -1.55000000000000004 < x < 1.55000000000000004

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.6%

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

Alternative 12: 60.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.8) (not (<= x 1.3))) (- 0.25 (/ 0.25 x)) (* 0.125 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.8) || !(x <= 1.3)) {
		tmp = 0.25 - (0.25 / x);
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.8d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = 0.25d0 - (0.25d0 / x)
    else
        tmp = 0.125d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.8) || !(x <= 1.3)) {
		tmp = 0.25 - (0.25 / x);
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.8) or not (x <= 1.3):
		tmp = 0.25 - (0.25 / x)
	else:
		tmp = 0.125 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.8) || !(x <= 1.3))
		tmp = Float64(0.25 - Float64(0.25 / x));
	else
		tmp = Float64(0.125 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.8) || ~((x <= 1.3)))
		tmp = 0.25 - (0.25 / x);
	else
		tmp = 0.125 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.8], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(0.25 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004 or 1.30000000000000004 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. metadata-eval 97.8%

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

   10. Simplified 97.8%

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

   11. Taylor expanded in x around 0 22.7%

       \[\leadsto \color{blue}{0.25 - 0.25 \cdot \frac{1}{x}} \]
   12. Step-by-step derivation

       1. associate-*r/ 22.7%

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

       2. metadata-eval 22.7%

          \[\leadsto 0.25 - \frac{\color{blue}{0.25}}{x} \]

   13. Simplified 22.7%

       \[\leadsto \color{blue}{0.25 - \frac{0.25}{x}} \]

   if -1.80000000000000004 < x < 1.30000000000000004

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 13: 58.6% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;0.18181818181818182\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   0.18181818181818182
   (if (<= x 1.2) (* 0.125 (* x x)) 0.18181818181818182)))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = 0.18181818181818182;
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.18181818181818182;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.2d0)) then
        tmp = 0.18181818181818182d0
    else if (x <= 1.2d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 0.18181818181818182d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = 0.18181818181818182;
	} else if (x <= 1.2) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.18181818181818182;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.2:
		tmp = 0.18181818181818182
	elif x <= 1.2:
		tmp = 0.125 * (x * x)
	else:
		tmp = 0.18181818181818182
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = 0.18181818181818182;
	elseif (x <= 1.2)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = 0.18181818181818182;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.2)
		tmp = 0.18181818181818182;
	elseif (x <= 1.2)
		tmp = 0.125 * (x * x);
	else
		tmp = 0.18181818181818182;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], 0.18181818181818182, If[LessEqual[x, 1.2], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.18181818181818182]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996 or 1.19999999999999996 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 19.5%

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

      1. associate-*r/ 19.5%

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

      2. metadata-eval 19.5%

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

      3. unpow2 19.5%

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

   10. Simplified 19.5%

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

   11. Taylor expanded in x around inf 19.5%

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

   if -1.19999999999999996 < x < 1.19999999999999996

   1. Initial program 51.1%

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

      1. distribute-lft-in 51.1%

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

      2. metadata-eval 51.1%

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

      3. associate-*r/ 51.1%

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

      4. metadata-eval 51.1%

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

   3. Simplified 51.1%

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

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 99.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;0.18181818181818182\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \]

Alternative 14: 58.3% accurate, 23.3× speedup

Math

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

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(8.0 / Float64(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[(8.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

   1. distribute-lft-in 74.8%

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

   2. metadata-eval 74.8%

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

   3. associate-*r/ 74.8%

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

   4. metadata-eval 74.8%

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

3. Simplified 74.8%

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

   1. flip-- 74.8%

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

   2. div-inv 74.8%

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

   3. metadata-eval 74.8%

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

   4. add-sqr-sqrt 75.5%

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

   5. associate--r+ 75.5%

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

   6. metadata-eval 75.5%

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

5. Applied egg-rr 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-/r/ 75.6%

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

7. Simplified 75.6%

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

8. Taylor expanded in x around 0 59.2%

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

   1. associate-*r/ 59.2%

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

   2. metadata-eval 59.2%

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

   3. unpow2 59.2%

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

10. Simplified 59.2%

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

11. Final simplification 59.2%

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

Alternative 15: 35.5% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-77}:\\ \;\;\;\;0.18181818181818182\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9e-77)
   0.18181818181818182
   (if (<= x 1.9e-77) 0.0 0.18181818181818182)))

C

double code(double x) {
	double tmp;
	if (x <= -1.9e-77) {
		tmp = 0.18181818181818182;
	} else if (x <= 1.9e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.18181818181818182;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.9d-77)) then
        tmp = 0.18181818181818182d0
    else if (x <= 1.9d-77) then
        tmp = 0.0d0
    else
        tmp = 0.18181818181818182d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.9e-77) {
		tmp = 0.18181818181818182;
	} else if (x <= 1.9e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.18181818181818182;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.9e-77:
		tmp = 0.18181818181818182
	elif x <= 1.9e-77:
		tmp = 0.0
	else:
		tmp = 0.18181818181818182
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9e-77)
		tmp = 0.18181818181818182;
	elseif (x <= 1.9e-77)
		tmp = 0.0;
	else
		tmp = 0.18181818181818182;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.9e-77)
		tmp = 0.18181818181818182;
	elseif (x <= 1.9e-77)
		tmp = 0.0;
	else
		tmp = 0.18181818181818182;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.9e-77], 0.18181818181818182, If[LessEqual[x, 1.9e-77], 0.0, 0.18181818181818182]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8999999999999999e-77 or 1.8999999999999999e-77 < x

   1. Initial program 80.6%

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

      1. distribute-lft-in 80.6%

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

      2. metadata-eval 80.6%

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

      3. associate-*r/ 80.6%

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

      4. metadata-eval 80.6%

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

   3. Simplified 80.6%

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

      1. flip-- 80.6%

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

      2. div-inv 80.6%

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

      3. metadata-eval 80.6%

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

      4. add-sqr-sqrt 81.8%

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

      5. associate--r+ 81.8%

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

      6. metadata-eval 81.8%

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

   5. Applied egg-rr 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-/r/ 81.8%

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

   7. Simplified 81.8%

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

   8. Taylor expanded in x around 0 35.2%

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

      1. associate-*r/ 35.2%

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

      2. metadata-eval 35.2%

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

      3. unpow2 35.2%

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

   10. Simplified 35.2%

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

   11. Taylor expanded in x around inf 17.1%

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

   if -1.8999999999999999e-77 < x < 1.8999999999999999e-77

   1. Initial program 65.3%

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

      1. distribute-lft-in 65.3%

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

      2. metadata-eval 65.3%

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

      3. associate-*r/ 65.3%

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

      4. metadata-eval 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in x around 0 65.3%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-77}:\\ \;\;\;\;0.18181818181818182\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.18181818181818182\\ \end{array} \]

Alternative 16: 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 74.8%

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

   1. distribute-lft-in 74.8%

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

   2. metadata-eval 74.8%

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

   3. associate-*r/ 74.8%

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

   4. metadata-eval 74.8%

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

3. Simplified 74.8%

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

   1. flip-- 74.8%

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

   2. div-inv 74.8%

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

   3. metadata-eval 74.8%

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

   4. add-sqr-sqrt 75.5%

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

   5. associate--r+ 75.5%

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

   6. metadata-eval 75.5%

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

5. Applied egg-rr 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-/r/ 75.6%

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

7. Simplified 75.6%

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

8. Taylor expanded in x around 0 59.2%

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

   1. associate-*r/ 59.2%

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

   2. metadata-eval 59.2%

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

   3. unpow2 59.2%

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

10. Simplified 59.2%

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

11. Taylor expanded in x around inf 11.9%

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

12. Final simplification 11.9%

    \[\leadsto 0.18181818181818182 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))