Given's Rotation SVD example, simplified

Percentage Accurate: 75.8% → 99.6%

Time: 7.2s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% 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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 2.0)
     (fma (* x 0.125) 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((x * 0.125), 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(Float64(x * 0.125), 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[(N[(x * 0.125), $MachinePrecision] * x + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. fma-def 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * 0.125), $MachinePrecision] * x + 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[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 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. fma-def 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around -inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 99.7%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. associate-*r* 100.0%

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

      2. fma-def 100.0%

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

   10. Applied egg-rr 100.0%

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

   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 97.8%

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

      1. associate-*r/ 97.8%

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

      2. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

      1. flip-- 97.8%

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

      2. div-inv 97.8%

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

      3. metadata-eval 97.8%

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

      4. add-sqr-sqrt 99.3%

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

      5. associate--r- 99.3%

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

      6. metadata-eval 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-frac 99.3%

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

      9. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.7%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

   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 97.8%

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

      1. associate-*r/ 97.8%

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

      2. metadata-eval 97.8%

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

   6. Simplified 97.8%

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

      1. flip-- 97.8%

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

      2. div-inv 97.8%

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

      3. metadata-eval 97.8%

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

      4. add-sqr-sqrt 99.3%

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

      5. associate--r- 99.3%

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

      6. metadata-eval 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-neg-frac 99.3%

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

      9. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.7%

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

Alternative 5: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 1.15)))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (+ (* 0.125 (* x x)) (* -0.0859375 (* (* x x) (* x x))))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.15)) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else {
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    else
        tmp = (0.125d0 * (x * x)) + ((-0.0859375d0) * ((x * x) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.15)) {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	} else {
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.15):
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	else:
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 1.15))
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(Float64(0.125 * Float64(x * x)) + Float64(-0.0859375 * Float64(Float64(x * x) * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 1.15)))
		tmp = 0.5 / (1.0 + sqrt(0.5));
	else
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 1.1499999999999999 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. flip-- 97.1%

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

      2. metadata-eval 97.1%

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

      3. add-sqr-sqrt 98.6%

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

      4. metadata-eval 98.6%

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

      5. div-inv 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   8. Simplified 98.6%

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

   if -1.1000000000000001 < x < 1.1499999999999999

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 1.15)))
   (- 1.0 (sqrt 0.5))
   (+ (* 0.125 (* x x)) (* -0.0859375 (* (* x x) (* x x))))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.15)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = (0.125d0 * (x * x)) + ((-0.0859375d0) * ((x * x) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.15)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.15):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 1.15))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(Float64(0.125 * Float64(x * x)) + Float64(-0.0859375 * Float64(Float64(x * x) * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 1.15)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 1.1499999999999999 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 97.1%

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

   if -1.1000000000000001 < x < 1.1499999999999999

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 7: 61.6% accurate, 11.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 1.15)))
   (+ 0.25 (/ 0.25 x))
   (+ (* 0.125 (* x x)) (* -0.0859375 (* (* x x) (* x x))))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = 0.25d0 + (0.25d0 / x)
    else
        tmp = (0.125d0 * (x * x)) + ((-0.0859375d0) * ((x * x) * (x * x)))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.15):
		tmp = 0.25 + (0.25 / x)
	else:
		tmp = (0.125 * (x * x)) + (-0.0859375 * ((x * x) * (x * x)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(0.25 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 1.1499999999999999 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around -inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 22.7%

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

       1. associate-*r/ 22.7%

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

       2. metadata-eval 22.7%

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

   11. Simplified 22.7%

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

   if -1.1000000000000001 < x < 1.1499999999999999

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. fma-def 100.0%

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

      2. unpow2 100.0%

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

   6. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   8. Applied egg-rr 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 60.5%

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

Alternative 8: 61.3% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1.8\right):\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.25) (not (<= x 1.8))) (+ 0.25 (/ 0.25 x)) (* 0.125 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.25) || !(x <= 1.8)) {
		tmp = 0.25 + (0.25 / x);
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.25d0)) .or. (.not. (x <= 1.8d0))) then
        tmp = 0.25d0 + (0.25d0 / x)
    else
        tmp = 0.125d0 * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.25) || !(x <= 1.8)) {
		tmp = 0.25 + (0.25 / x);
	} else {
		tmp = 0.125 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.25) or not (x <= 1.8):
		tmp = 0.25 + (0.25 / x)
	else:
		tmp = 0.125 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.25) || !(x <= 1.8))
		tmp = Float64(0.25 + Float64(0.25 / x));
	else
		tmp = Float64(0.125 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.25) || ~((x <= 1.8)))
		tmp = 0.25 + (0.25 / x);
	else
		tmp = 0.125 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.25], N[Not[LessEqual[x, 1.8]], $MachinePrecision]], N[(0.25 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25 or 1.80000000000000004 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around -inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 22.7%

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

       1. associate-*r/ 22.7%

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

       2. metadata-eval 22.7%

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

   11. Simplified 22.7%

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

   if -1.25 < x < 1.80000000000000004

   1. Initial program 55.5%

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

      1. distribute-lft-in 55.5%

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

      2. metadata-eval 55.5%

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

      3. associate-*r/ 55.5%

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

      4. metadata-eval 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. unpow2 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \lor \neg \left(x \leq 1.8\right):\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 9: 52.2% 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 77.5%

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

   1. distribute-lft-in 77.5%

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

   2. metadata-eval 77.5%

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

   3. associate-*r/ 77.5%

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

   4. metadata-eval 77.5%

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

3. Simplified 77.5%

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

4. Taylor expanded in x around 0 50.9%

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

   1. unpow2 50.9%

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

6. Simplified 50.9%

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

7. Final simplification 50.9%

   \[\leadsto 0.125 \cdot \left(x \cdot x\right) \]

Alternative 10: 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 77.5%

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

   1. distribute-lft-in 77.5%

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

   2. metadata-eval 77.5%

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

   3. associate-*r/ 77.5%

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

   4. metadata-eval 77.5%

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

3. Simplified 77.5%

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

4. Taylor expanded in x around 0 28.6%

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

5. Final simplification 28.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023178 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))