Given's Rotation SVD example, simplified

Percentage Accurate: 75.2% → 99.6%

Time: 5.9s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.2% 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: 99.6% 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 #s(literal 1 binary64) x) < 2

   1. Initial program 54.8%

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

      1. distribute-lft-in 54.8%

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

      2. metadata-eval 54.8%

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

      3. associate-*r/ 54.8%

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

      4. metadata-eval 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. fma-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-sqr 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. 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. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (fma 0.125 (* x x) (* -0.0859375 (pow x 4.0)))
   (* (- 0.5 (/ 0.5 x)) (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 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 = (0.5 - (0.5 / x)) * (1.0 / (1.0 + sqrt((0.5 + (0.5 / 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(Float64(0.5 - Float64(0.5 / x)) * Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / 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[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.8%

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

      1. distribute-lft-in 54.8%

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

      2. metadata-eval 54.8%

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

      3. associate-*r/ 54.8%

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

      4. metadata-eval 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 100.0%

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

      2. fma-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-sqr 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 97.4%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 + 0.5 \cdot \frac{1}{x}}} \]
   6. 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%

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

   7. Simplified 97.4%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5}{x}}} \]
   8. 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 98.9%

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

      5. associate--r+ 98.9%

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

      6. metadata-eval 98.9%

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

   9. Applied egg-rr 98.9%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

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

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 (+ 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 / (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 / (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 / (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(Float64(x * x) * -0.0859375)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) * Float64(1.0 / 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 / (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[(N[(x * x), $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.8%

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

      1. distribute-lft-in 54.8%

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

      2. metadata-eval 54.8%

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

      3. associate-*r/ 54.8%

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

      4. metadata-eval 54.8%

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

   3. Simplified 54.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot -0.0859375\right) \]
   10. Step-by-step derivation

       1. unpow2 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 97.4%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 + 0.5 \cdot \frac{1}{x}}} \]
   6. 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%

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

   7. Simplified 97.4%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 + \frac{0.5}{x}}} \]
   8. 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 98.9%

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

      5. associate--r+ 98.9%

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

      6. metadata-eval 98.9%

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

   9. Applied egg-rr 98.9%

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

Alternative 4: 74.6% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (* (* x x) (+ 0.125 (* (* x x) -0.0859375)))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d0) then
        tmp = (x * x) * (0.125d0 + ((x * x) * (-0.0859375d0)))
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1) {
		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 x <= 1.1:
		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 (x <= 1.1)
		tmp = Float64(Float64(x * x) * Float64(0.125 + Float64(Float64(x * 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 (x <= 1.1)
		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[x, 1.1], N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(N[(x * x), $MachinePrecision] * -0.0859375), $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 x < 1.1000000000000001

   1. Initial program 68.1%

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

      1. distribute-lft-in 68.1%

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

      2. metadata-eval 68.1%

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

      3. associate-*r/ 68.1%

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

      4. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 69.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 69.8%

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

   7. Simplified 69.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)} \]
   8. Step-by-step derivation

      1. unpow2 69.7%

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

   9. Applied egg-rr 69.8%

      \[\leadsto {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot -0.0859375\right) \]
   10. Step-by-step derivation

       1. unpow2 69.7%

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

   11. Applied egg-rr 69.8%

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

   if 1.1000000000000001 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.7%

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

Alternative 5: 74.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (* (* x x) (+ 0.125 (* (* x x) -0.0859375)))
   (- 1.0 (sqrt 0.5))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d0) then
        tmp = (x * x) * (0.125d0 + ((x * x) * (-0.0859375d0)))
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.1:
		tmp = (x * x) * (0.125 + ((x * x) * -0.0859375))
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.1], N[(N[(x * x), $MachinePrecision] * N[(0.125 + N[(N[(x * x), $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 68.1%

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

      1. distribute-lft-in 68.1%

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

      2. metadata-eval 68.1%

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

      3. associate-*r/ 68.1%

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

      4. metadata-eval 68.1%

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

   3. Simplified 68.1%

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

   5. Taylor expanded in x around 0 69.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + -0.0859375 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 69.8%

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

   7. Simplified 69.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)} \]
   8. Step-by-step derivation

      1. unpow2 69.7%

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

   9. Applied egg-rr 69.8%

      \[\leadsto {x}^{2} \cdot \left(0.125 + \color{blue}{\left(x \cdot x\right)} \cdot -0.0859375\right) \]
   10. Step-by-step derivation

       1. unpow2 69.7%

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

   11. Applied egg-rr 69.8%

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

   if 1.1000000000000001 < x

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in x around inf 97.2%

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

Alternative 6: 51.7% accurate, 42.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.125 (* x x)))

C

double code(double x) {
	return 0.125 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.125d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.125 * (x * x);
}

Python

def code(x):
	return 0.125 * (x * x)

Julia

function code(x)
	return Float64(0.125 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.125 * (x * x);
end

Wolfram

code[x_] := N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.4%

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

   1. distribute-lft-in 75.4%

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

   2. metadata-eval 75.4%

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

   3. associate-*r/ 75.4%

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

   4. metadata-eval 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in x around 0 54.7%

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

   1. unpow2 52.9%

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

7. Applied egg-rr 54.7%

   \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]
8. Add Preprocessing

Alternative 7: 26.9% 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 75.4%

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

   1. distribute-lft-in 75.4%

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

   2. metadata-eval 75.4%

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

   3. associate-*r/ 75.4%

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

   4. metadata-eval 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in x around 0 30.2%

   \[\leadsto 1 - \color{blue}{1} \]
6. Step-by-step derivation

   1. metadata-eval 30.2%

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 30.2%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))