Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 99.8%

Time: 12.4s

Alternatives: 11

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: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. /-rgt-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

4. Final simplification 99.2%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(x * x) * Float64(0.125 + Float64(x * Float64(x * -0.0859375))));
	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 = (x * x) * (0.125 + (x * (x * -0.0859375)));
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   5. Applied egg-rr 64.0%

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

      1. associate-*r/ 64.0%

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

      2. *-rgt-identity 64.0%

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

      3. /-rgt-identity 64.0%

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

      4. sub-neg 64.0%

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

      5. distribute-neg-frac 64.0%

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

      6. metadata-eval 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. unpow2 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.9%

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

       2. metadata-eval 99.9%

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

       3. pow-prod-up 99.9%

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

       4. pow2 99.9%

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

       5. pow2 99.9%

          \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125 \]

       6. associate-*l* 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375\right)} + \left(x \cdot x\right) \cdot 0.125 \]

       7. distribute-lft-out 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375 + 0.125\right)} \]

       8. associate-*l* 99.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot -0.0859375\right)} + 0.125\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.0859375\right) + 0.125\right)} \]

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

4. Final simplification 99.2%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   5. Applied egg-rr 64.0%

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

      1. associate-*r/ 64.0%

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

      2. *-rgt-identity 64.0%

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

      3. /-rgt-identity 64.0%

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

      4. sub-neg 64.0%

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

      5. distribute-neg-frac 64.0%

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

      6. metadata-eval 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. unpow2 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.9%

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

       2. metadata-eval 99.9%

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

       3. pow-prod-up 99.9%

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

       4. pow2 99.9%

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

       5. pow2 99.9%

          \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125 \]

       6. associate-*l* 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375\right)} + \left(x \cdot x\right) \cdot 0.125 \]

       7. distribute-lft-out 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375 + 0.125\right)} \]

       8. associate-*l* 99.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot -0.0859375\right)} + 0.125\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.0859375\right) + 0.125\right)} \]

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 96.7%

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

      1. flip-- 96.7%

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

      2. frac-2neg 96.7%

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

   6. Applied egg-rr 98.1%

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

      1. associate-+r+ 98.1%

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

      2. metadata-eval 98.1%

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

      3. distribute-neg-in 98.1%

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

      4. metadata-eval 98.1%

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

      5. metadata-eval 98.1%

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

      6. *-inverses 98.1%

         \[\leadsto \frac{\color{blue}{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}}} \cdot -0.5 + \left(-\frac{-0.5}{x}\right)}{-\left(1 + \sqrt{0.5 + \frac{0.5}{x}}\right)} \]

      7. distribute-neg-frac 98.1%

         \[\leadsto \frac{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}} \cdot -0.5 + \color{blue}{\frac{--0.5}{x}}}{-\left(1 + \sqrt{0.5 + \frac{0.5}{x}}\right)} \]

      8. metadata-eval 98.1%

         \[\leadsto \frac{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}} \cdot -0.5 + \frac{\color{blue}{0.5}}{x}}{-\left(1 + \sqrt{0.5 + \frac{0.5}{x}}\right)} \]

      9. *-lft-identity 98.1%

         \[\leadsto \frac{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}} \cdot -0.5 + \color{blue}{1 \cdot \frac{0.5}{x}}}{-\left(1 + \sqrt{0.5 + \frac{0.5}{x}}\right)} \]

      10. *-inverses 98.1%

          \[\leadsto \frac{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}} \cdot -0.5 + \color{blue}{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}}} \cdot \frac{0.5}{x}}{-\left(1 + \sqrt{0.5 + \frac{0.5}{x}}\right)} \]

      11. distribute-lft-in 98.1%

          \[\leadsto \frac{\color{blue}{\frac{0.25 + -0.25 \cdot {x}^{-2}}{0.25 + -0.25 \cdot {x}^{-2}} \cdot \left(-0.5 + \frac{0.5}{x}\right)}}{-\left(1 + \sqrt{0.5 + \frac{0.5}{x}}\right)} \]

      12. *-inverses 98.1%

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

      13. *-lft-identity 98.1%

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

      14. +-commutative 98.1%

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

      15. distribute-neg-in 98.1%

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

      16. metadata-eval 98.1%

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

      17. unsub-neg 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 98.9%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   5. Applied egg-rr 64.0%

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

      1. associate-*r/ 64.0%

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

      2. *-rgt-identity 64.0%

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

      3. /-rgt-identity 64.0%

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

      4. sub-neg 64.0%

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

      5. distribute-neg-frac 64.0%

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

      6. metadata-eval 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. unpow2 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.9%

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

       2. metadata-eval 99.9%

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

       3. pow-prod-up 99.9%

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

       4. pow2 99.9%

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

       5. pow2 99.9%

          \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125 \]

       6. associate-*l* 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375\right)} + \left(x \cdot x\right) \cdot 0.125 \]

       7. distribute-lft-out 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375 + 0.125\right)} \]

       8. associate-*l* 99.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot -0.0859375\right)} + 0.125\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.0859375\right) + 0.125\right)} \]

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. /-rgt-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 96.8%

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

4. Final simplification 98.2%

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

Alternative 6: 98.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 1.1)))
   (- 1.0 (sqrt 0.5))
   (* (* x x) (+ 0.125 (* x (* x -0.0859375))))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.1)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.1d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = (x * x) * (0.125d0 + (x * (x * (-0.0859375d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.1)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.1):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 1.1))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(Float64(x * x) * Float64(0.125 + Float64(x * Float64(x * -0.0859375))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 1.1)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.1]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(x * N[(x * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 1.1000000000000001 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 95.5%

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

   if -1.1000000000000001 < x < 1.1000000000000001

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   5. Applied egg-rr 64.0%

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

      1. associate-*r/ 64.0%

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

      2. *-rgt-identity 64.0%

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

      3. /-rgt-identity 64.0%

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

      4. sub-neg 64.0%

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

      5. distribute-neg-frac 64.0%

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

      6. metadata-eval 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. unpow2 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.9%

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

       2. metadata-eval 99.9%

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

       3. pow-prod-up 99.9%

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

       4. pow2 99.9%

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

       5. pow2 99.9%

          \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125 \]

       6. associate-*l* 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375\right)} + \left(x \cdot x\right) \cdot 0.125 \]

       7. distribute-lft-out 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375 + 0.125\right)} \]

       8. associate-*l* 99.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot -0.0859375\right)} + 0.125\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.0859375\right) + 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

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

Alternative 7: 61.1% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{2}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + x \cdot \left(x \cdot -0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (/ (- 0.5 (/ 0.5 x)) 2.0)
   (if (<= x 1.1) (* (* x x) (+ 0.125 (* x (* x -0.0859375)))) 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (0.5 - (0.5 / x)) / 2.0;
	} else if (x <= 1.1) {
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)));
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.15d0)) then
        tmp = (0.5d0 - (0.5d0 / x)) / 2.0d0
    else if (x <= 1.1d0) then
        tmp = (x * x) * (0.125d0 + (x * (x * (-0.0859375d0))))
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (0.5 - (0.5 / x)) / 2.0;
	} else if (x <= 1.1) {
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)));
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = (0.5 - (0.5 / x)) / 2.0
	elif x <= 1.1:
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)))
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / 2.0);
	elseif (x <= 1.1)
		tmp = Float64(Float64(x * x) * Float64(0.125 + Float64(x * Float64(x * -0.0859375))));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = (0.5 - (0.5 / x)) / 2.0;
	elseif (x <= 1.1)
		tmp = (x * x) * (0.125 + (x * (x * -0.0859375)));
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.1], N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(x * N[(x * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.25]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   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%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. /-rgt-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 22.7%

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

   9. Taylor expanded in x around inf 22.7%

      \[\leadsto \frac{\color{blue}{0.5 - 0.5 \cdot \frac{1}{x}}}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.7%

          \[\leadsto \frac{0.5 - \color{blue}{\frac{0.5 \cdot 1}{x}}}{2} \]

       2. metadata-eval 22.7%

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

   11. Simplified 22.7%

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

   if -1.1499999999999999 < x < 1.1000000000000001

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   5. Applied egg-rr 64.0%

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

      1. associate-*r/ 64.0%

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

      2. *-rgt-identity 64.0%

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

      3. /-rgt-identity 64.0%

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

      4. sub-neg 64.0%

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

      5. distribute-neg-frac 64.0%

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

      6. metadata-eval 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in x around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. *-commutative 99.9%

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

      4. unpow2 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, -0.0859375, \left(x \cdot x\right) \cdot 0.125\right)} \]
   11. Step-by-step derivation

       1. fma-udef 99.9%

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

       2. metadata-eval 99.9%

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

       3. pow-prod-up 99.9%

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

       4. pow2 99.9%

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

       5. pow2 99.9%

          \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.0859375 + \left(x \cdot x\right) \cdot 0.125 \]

       6. associate-*l* 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375\right)} + \left(x \cdot x\right) \cdot 0.125 \]

       7. distribute-lft-out 99.9%

          \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0859375 + 0.125\right)} \]

       8. associate-*l* 99.9%

          \[\leadsto \left(x \cdot x\right) \cdot \left(\color{blue}{x \cdot \left(x \cdot -0.0859375\right)} + 0.125\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.0859375\right) + 0.125\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. /-rgt-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 22.7%

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

   9. Taylor expanded in x around inf 22.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{2}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + x \cdot \left(x \cdot -0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 8: 60.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4) 0.25 (if (<= x 1.45) (* 0.125 (* x x)) 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = 0.25;
	} else if (x <= 1.45) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.4d0)) then
        tmp = 0.25d0
    else if (x <= 1.45d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = 0.25;
	} else if (x <= 1.45) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.4:
		tmp = 0.25
	elif x <= 1.45:
		tmp = 0.125 * (x * x)
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = 0.25;
	elseif (x <= 1.45)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = 0.25;
	elseif (x <= 1.45)
		tmp = 0.125 * (x * x);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], 0.25, If[LessEqual[x, 1.45], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.44999999999999996 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. /-rgt-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 22.7%

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

   9. Taylor expanded in x around inf 22.8%

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

   if -1.3999999999999999 < x < 1.44999999999999996

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 9: 60.9% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.78:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.78)
   (/ (- 0.5 (/ 0.5 x)) 2.0)
   (if (<= x 1.45) (* 0.125 (* x x)) 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -1.78) {
		tmp = (0.5 - (0.5 / x)) / 2.0;
	} else if (x <= 1.45) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.78d0)) then
        tmp = (0.5d0 - (0.5d0 / x)) / 2.0d0
    else if (x <= 1.45d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.78) {
		tmp = (0.5 - (0.5 / x)) / 2.0;
	} else if (x <= 1.45) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.78:
		tmp = (0.5 - (0.5 / x)) / 2.0
	elif x <= 1.45:
		tmp = 0.125 * (x * x)
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.78)
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / 2.0);
	elseif (x <= 1.45)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.78)
		tmp = (0.5 - (0.5 / x)) / 2.0;
	elseif (x <= 1.45)
		tmp = 0.125 * (x * x);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.78], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.45], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.25]]

Derivation

1. Split input into 3 regimes

2. if x < -1.78000000000000003

   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%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. /-rgt-identity 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 22.7%

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

   9. Taylor expanded in x around inf 22.7%

      \[\leadsto \frac{\color{blue}{0.5 - 0.5 \cdot \frac{1}{x}}}{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 22.7%

          \[\leadsto \frac{0.5 - \color{blue}{\frac{0.5 \cdot 1}{x}}}{2} \]

       2. metadata-eval 22.7%

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

   11. Simplified 22.7%

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

   if -1.78000000000000003 < x < 1.44999999999999996

   1. Initial program 63.9%

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

      1. distribute-lft-in 63.9%

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

      2. metadata-eval 63.9%

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

      3. associate-*r/ 63.9%

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

      4. metadata-eval 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in x around 0 99.6%

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

      1. unpow2 99.6%

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

   6. Simplified 99.6%

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

   if 1.44999999999999996 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. /-rgt-identity 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 22.7%

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

   9. Taylor expanded in x around inf 22.9%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.78:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 10: 38.1% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.4%

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

      1. distribute-lft-in 86.4%

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

      2. metadata-eval 86.4%

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

      3. associate-*r/ 86.4%

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

      4. metadata-eval 86.4%

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

   3. Simplified 86.4%

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

   5. Applied egg-rr 87.6%

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

      1. associate-*r/ 87.6%

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

      2. *-rgt-identity 87.6%

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

      3. /-rgt-identity 87.6%

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

      4. sub-neg 87.6%

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

      5. distribute-neg-frac 87.6%

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

      6. metadata-eval 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in x around 0 20.6%

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

   9. Taylor expanded in x around inf 20.8%

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

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

   1. Initial program 76.4%

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

      1. distribute-lft-in 76.4%

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

      2. metadata-eval 76.4%

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

      3. associate-*r/ 76.4%

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

      4. metadata-eval 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in x around 0 76.4%

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

4. Final simplification 41.8%

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

Alternative 11: 27.9% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 82.6%

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

   1. distribute-lft-in 82.6%

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

   2. metadata-eval 82.6%

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

   3. associate-*r/ 82.6%

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

   4. metadata-eval 82.6%

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

3. Simplified 82.6%

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

4. Taylor expanded in x around 0 30.9%

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

5. Final simplification 30.9%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))