Given's Rotation SVD example, simplified

Percentage Accurate: 75.8% → 99.9%

Time: 12.5s

Alternatives: 14

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: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{1 + x \cdot x} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (fma (* x 0.125) x (* -0.0859375 (pow x 4.0)))
   (/
    (*
     (/
      (- 0.25 (pow (/ 0.015625 (pow (hypot 1.0 x) 6.0)) 0.3333333333333333))
      (+ 0.125 (/ 0.125 (pow (hypot 1.0 x) 3.0))))
     (+ 0.25 (- (/ 0.25 (+ 1.0 (* x x))) (/ 0.25 (hypot 1.0 x)))))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma((x * 0.125), x, (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = (((0.25 - pow((0.015625 / pow(hypot(1.0, x), 6.0)), 0.3333333333333333)) / (0.125 + (0.125 / pow(hypot(1.0, x), 3.0)))) * (0.25 + ((0.25 / (1.0 + (x * x))) - (0.25 / hypot(1.0, 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) <= 1.0002)
		tmp = fma(Float64(x * 0.125), x, Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(Float64(Float64(0.25 - (Float64(0.015625 / (hypot(1.0, x) ^ 6.0)) ^ 0.3333333333333333)) / Float64(0.125 + Float64(0.125 / (hypot(1.0, x) ^ 3.0)))) * Float64(0.25 + Float64(Float64(0.25 / Float64(1.0 + Float64(x * x))) - Float64(0.25 / hypot(1.0, 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], 1.0002], N[(N[(x * 0.125), $MachinePrecision] * x + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.25 - N[Power[N[(0.015625 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision] / N[(0.125 + N[(0.125 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.25 + N[(N[(0.25 / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. metadata-eval 48.1%

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

      3. associate-*r/ 48.1%

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

      4. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]
   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 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. flip-- 99.8%

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

      2. flip3-+ 99.8%

         \[\leadsto \frac{\frac{0.5 \cdot 0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{\frac{{0.5}^{3} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. associate-/r/ 99.8%

         \[\leadsto \frac{\color{blue}{\frac{0.5 \cdot 0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{{0.5}^{3} + {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}} \cdot \left(0.5 \cdot 0.5 + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(\left(0.25 + \frac{0.25}{1 + x \cdot x}\right) - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   8. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{1 + \color{blue}{{x}^{2}}}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(\left(0.25 + \frac{0.25}{1 + x \cdot x}\right) - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. +-commutative 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{\color{blue}{{x}^{2} + 1}}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(\left(0.25 + \frac{0.25}{1 + x \cdot x}\right) - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. unpow2 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{\color{blue}{x \cdot x} + 1}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(\left(0.25 + \frac{0.25}{1 + x \cdot x}\right) - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. associate--l+ 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \color{blue}{\left(0.25 + \left(\frac{0.25}{1 + x \cdot x} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. unpow2 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{1 + \color{blue}{{x}^{2}}} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. +-commutative 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{\color{blue}{{x}^{2} + 1}} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      7. unpow2 99.8%

         \[\leadsto \frac{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{\color{blue}{x \cdot x} + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   9. Simplified 99.8%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   10. Step-by-step derivation

       1. add-cbrt-cube 99.8%

          \[\leadsto \frac{\frac{0.25 - \color{blue}{\sqrt[3]{\left(\frac{0.25}{x \cdot x + 1} \cdot \frac{0.25}{x \cdot x + 1}\right) \cdot \frac{0.25}{x \cdot x + 1}}}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       2. pow1/3 99.9%

          \[\leadsto \frac{\frac{0.25 - \color{blue}{{\left(\left(\frac{0.25}{x \cdot x + 1} \cdot \frac{0.25}{x \cdot x + 1}\right) \cdot \frac{0.25}{x \cdot x + 1}\right)}^{0.3333333333333333}}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. pow3 99.9%

          \[\leadsto \frac{\frac{0.25 - {\color{blue}{\left({\left(\frac{0.25}{x \cdot x + 1}\right)}^{3}\right)}}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       4. cube-div 99.9%

          \[\leadsto \frac{\frac{0.25 - {\color{blue}{\left(\frac{{0.25}^{3}}{{\left(x \cdot x + 1\right)}^{3}}\right)}}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       5. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{\color{blue}{0.015625}}{{\left(x \cdot x + 1\right)}^{3}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       6. add-sqr-sqrt 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\color{blue}{\left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}\right)}}^{3}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       7. +-commutative 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\sqrt{\color{blue}{1 + x \cdot x}} \cdot \sqrt{x \cdot x + 1}\right)}^{3}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       8. hypot-1-def 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\color{blue}{\mathsf{hypot}\left(1, x\right)} \cdot \sqrt{x \cdot x + 1}\right)}^{3}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       9. +-commutative 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right) \cdot \sqrt{\color{blue}{1 + x \cdot x}}\right)}^{3}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       10. hypot-1-def 99.9%

           \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right) \cdot \color{blue}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       11. pow-prod-down 99.9%

           \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{\color{blue}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3} \cdot {\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       12. pow-prod-up 99.9%

           \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{\color{blue}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{\left(3 + 3\right)}}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       13. metadata-eval 99.9%

           \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{6}}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{0.25 - \color{blue}{{\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{0.3333333333333333}}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{x \cdot x + 1} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{0.3333333333333333}}{0.125 + \frac{0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}} \cdot \left(0.25 + \left(\frac{0.25}{1 + x \cdot x} - \frac{0.25}{\mathsf{hypot}\left(1, x\right)}\right)\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - {\left(\frac{0.000244140625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{12}}\right)}^{0.16666666666666666}}{t_0}}{1 + \sqrt{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0002)
     (fma (* x 0.125) x (* -0.0859375 (pow x 4.0)))
     (/
      (/
       (-
        0.25
        (pow (/ 0.000244140625 (pow (hypot 1.0 x) 12.0)) 0.16666666666666666))
       t_0)
      (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma((x * 0.125), x, (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = ((0.25 - pow((0.000244140625 / pow(hypot(1.0, x), 12.0)), 0.16666666666666666)) / t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = fma(Float64(x * 0.125), x, Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(Float64(0.25 - (Float64(0.000244140625 / (hypot(1.0, x) ^ 12.0)) ^ 0.16666666666666666)) / t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(x * 0.125), $MachinePrecision] * x + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 - N[Power[N[(0.000244140625 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 12.0], $MachinePrecision]), $MachinePrecision], 0.16666666666666666], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. metadata-eval 48.1%

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

      3. associate-*r/ 48.1%

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

      4. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]
   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 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. flip-- 99.8%

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

      2. metadata-eval 99.8%

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

      3. frac-times 99.8%

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

      4. metadata-eval 99.8%

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

      5. hypot-udef 99.8%

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

      6. hypot-udef 99.8%

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

      7. add-sqr-sqrt 99.8%

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

      8. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 99.8%

         \[\leadsto \frac{\frac{0.25 - \color{blue}{\sqrt[3]{\left(\frac{0.25}{1 + x \cdot x} \cdot \frac{0.25}{1 + x \cdot x}\right) \cdot \frac{0.25}{1 + x \cdot x}}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. pow1/3 99.9%

         \[\leadsto \frac{\frac{0.25 - \color{blue}{{\left(\left(\frac{0.25}{1 + x \cdot x} \cdot \frac{0.25}{1 + x \cdot x}\right) \cdot \frac{0.25}{1 + x \cdot x}\right)}^{0.3333333333333333}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. pow3 99.9%

         \[\leadsto \frac{\frac{0.25 - {\color{blue}{\left({\left(\frac{0.25}{1 + x \cdot x}\right)}^{3}\right)}}^{0.3333333333333333}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. cube-div 99.9%

         \[\leadsto \frac{\frac{0.25 - {\color{blue}{\left(\frac{{0.25}^{3}}{{\left(1 + x \cdot x\right)}^{3}}\right)}}^{0.3333333333333333}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{\frac{0.25 - {\left(\frac{\color{blue}{0.015625}}{{\left(1 + x \cdot x\right)}^{3}}\right)}^{0.3333333333333333}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\left(1 + x \cdot x\right)}^{\color{blue}{\left(\frac{6}{2}\right)}}}\right)}^{0.3333333333333333}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      7. sqrt-pow2 99.9%

         \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{\color{blue}{{\left(\sqrt{1 + x \cdot x}\right)}^{6}}}\right)}^{0.3333333333333333}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      8. hypot-1-def 99.9%

         \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.015625}{{\color{blue}{\left(\mathsf{hypot}\left(1, x\right)\right)}}^{6}}\right)}^{0.3333333333333333}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      9. sqr-pow 99.8%

         \[\leadsto \frac{\frac{0.25 - \color{blue}{{\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      10. pow-prod-down 99.9%

          \[\leadsto \frac{\frac{0.25 - \color{blue}{{\left(\frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}} \cdot \frac{0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      11. frac-times 99.9%

          \[\leadsto \frac{\frac{0.25 - {\color{blue}{\left(\frac{0.015625 \cdot 0.015625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6} \cdot {\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      12. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{\color{blue}{0.000244140625}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{6} \cdot {\left(\mathsf{hypot}\left(1, x\right)\right)}^{6}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      13. pow-prod-up 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.000244140625}{\color{blue}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{\left(6 + 6\right)}}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      14. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.000244140625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{12}}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      15. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.25 - {\left(\frac{0.000244140625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{12}}\right)}^{\color{blue}{0.16666666666666666}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   9. Applied egg-rr 99.9%

      \[\leadsto \frac{\frac{0.25 - \color{blue}{{\left(\frac{0.000244140625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{12}}\right)}^{0.16666666666666666}}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - {\left(\frac{0.000244140625}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{12}}\right)}^{0.16666666666666666}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \sqrt[3]{\frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (fma (* x 0.125) x (* -0.0859375 (pow x 4.0)))
   (*
    (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))
    (+ 0.5 (cbrt (/ -0.125 (pow (hypot 1.0 x) 3.0)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma((x * 0.125), x, (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = (1.0 / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))))) * (0.5 + cbrt((-0.125 / pow(hypot(1.0, x), 3.0))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = fma(Float64(x * 0.125), x, Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))))) * Float64(0.5 + cbrt(Float64(-0.125 / (hypot(1.0, x) ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(x * 0.125), $MachinePrecision] * x + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[Power[N[(-0.125 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. metadata-eval 48.1%

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

      3. associate-*r/ 48.1%

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

      4. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]
   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 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. add-cbrt-cube 99.9%

         \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \color{blue}{\sqrt[3]{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}\right) \]

      2. pow1/3 63.1%

         \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \color{blue}{{\left(\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.3333333333333333}}\right) \]

      3. pow3 63.1%

         \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + {\color{blue}{\left({\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. cube-div 63.9%

         \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + {\color{blue}{\left(\frac{{-0.5}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}\right)}}^{0.3333333333333333}\right) \]

      5. metadata-eval 63.9%

         \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + {\left(\frac{\color{blue}{-0.125}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}\right)}^{0.3333333333333333}\right) \]

   9. Applied egg-rr 63.9%

      \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \color{blue}{{\left(\frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}\right)}^{0.3333333333333333}}\right) \]
   10. Step-by-step derivation

       1. unpow1/3 99.9%

          \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \color{blue}{\sqrt[3]{\frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}\right) \]

   11. Simplified 99.9%

       \[\leadsto \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \color{blue}{\sqrt[3]{\frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \sqrt[3]{\frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}\right)\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0002)
     (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) <= 1.0002) {
		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) <= 1.0002)
		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], 1.0002], 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) < 1.0002

   1. Initial program 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. metadata-eval 48.1%

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

      3. associate-*r/ 48.1%

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

      4. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]
   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 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (* x x)))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * (x * x));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * (x * x));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * (x * x))
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * Float64(x * x)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x * x));
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. metadata-eval 48.1%

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

      3. associate-*r/ 48.1%

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

      4. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

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

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 6: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (fma (* x 0.125) x (* -0.0859375 (pow x 4.0)))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma((x * 0.125), x, (-0.0859375 * pow(x, 4.0)));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = fma(Float64(x * 0.125), x, Float64(-0.0859375 * (x ^ 4.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(x * 0.125), $MachinePrecision] * x + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.0002

   1. Initial program 48.1%

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

      1. distribute-lft-in 48.1%

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

      2. metadata-eval 48.1%

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

      3. associate-*r/ 48.1%

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

      4. metadata-eval 48.1%

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

   3. Simplified 48.1%

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

   4. Taylor expanded in x around 0 99.9%

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

      1. fma-def 99.9%

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

      2. unpow2 99.9%

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

   6. Simplified 99.9%

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

      1. fma-udef 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right) + -0.0859375 \cdot {x}^{4}} \]
   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 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 7: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[(x * x), $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 48.7%

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

      1. distribute-lft-in 48.7%

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

      2. metadata-eval 48.7%

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

      3. associate-*r/ 48.7%

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

      4. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. fma-def 99.1%

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

      2. unpow2 99.1%

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

   6. Simplified 99.1%

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

      1. fma-udef 99.1%

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

   8. Applied egg-rr 99.1%

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

   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 97.5%

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

      1. associate-*r/ 97.5%

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

      2. metadata-eval 97.5%

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

   6. Simplified 97.5%

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

      1. flip-- 97.5%

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

      2. div-inv 97.5%

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

      3. metadata-eval 97.5%

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

      4. add-sqr-sqrt 99.0%

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

      5. associate--r- 99.0%

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

      6. metadata-eval 99.0%

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

      7. sub-neg 99.0%

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

      8. distribute-neg-frac 99.0%

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

      9. metadata-eval 99.0%

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

   8. Applied egg-rr 99.0%

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

      1. associate-*r/ 99.0%

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

      2. *-rgt-identity 99.0%

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

   10. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 8: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

      1. distribute-lft-in 48.7%

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

      2. metadata-eval 48.7%

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

      3. associate-*r/ 48.7%

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

      4. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in x around 0 99.1%

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

      1. fma-def 99.1%

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

      2. unpow2 99.1%

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

   6. Simplified 99.1%

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

      1. fma-udef 99.1%

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

   8. Applied egg-rr 99.1%

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

   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 98.1%

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

4. Final simplification 98.6%

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

Alternative 9: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

      1. distribute-lft-in 48.7%

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

      2. metadata-eval 48.7%

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

      3. associate-*r/ 48.7%

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

      4. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

      1. flip-- 48.7%

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

      2. metadata-eval 48.7%

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

      3. add-sqr-sqrt 48.8%

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

      4. associate--r+ 48.8%

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

      5. metadata-eval 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in x around 0 98.1%

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

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot 0.125} \]

      2. unpow2 98.1%

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

      3. associate-*l* 98.1%

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

   8. Simplified 98.1%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. 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 98.1%

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

4. Final simplification 98.1%

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

Alternative 10: 97.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.55) (not (<= x 1.5))) (- 1.0 (sqrt 0.5)) (* x (* x 0.125))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.5)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.55d0)) .or. (.not. (x <= 1.5d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = x * (x * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.55) || !(x <= 1.5)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.55) or not (x <= 1.5):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = x * (x * 0.125)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.55) || !(x <= 1.5))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(x * Float64(x * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.55) || ~((x <= 1.5)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = x * (x * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.55], N[Not[LessEqual[x, 1.5]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 or 1.5 < x

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 96.6%

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

   if -1.55000000000000004 < x < 1.5

   1. Initial program 48.7%

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

      1. distribute-lft-in 48.7%

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

      2. metadata-eval 48.7%

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

      3. associate-*r/ 48.7%

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

      4. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

      1. flip-- 48.7%

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

      2. metadata-eval 48.7%

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

      3. add-sqr-sqrt 48.8%

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

      4. associate--r+ 48.8%

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

      5. metadata-eval 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in x around 0 98.1%

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

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot 0.125} \]

      2. unpow2 98.1%

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

      3. associate-*l* 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.5\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 11: 61.1% accurate, 23.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.25) || !(x <= 1.8)) {
		tmp = 0.25 + (0.25 / x);
	} else {
		tmp = x * (x * 0.125);
	}
	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 = x * (x * 0.125d0)
    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 = x * (x * 0.125);
	}
	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 = x * (x * 0.125)
	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(x * Float64(x * 0.125));
	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 = x * (x * 0.125);
	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[(x * N[(x * 0.125), $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 0 22.7%

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

   7. Taylor expanded in x around -inf 22.7%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot \frac{1}{x}} \]
   8. 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%

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

   9. Simplified 22.7%

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

   if -1.25 < x < 1.80000000000000004

   1. Initial program 48.7%

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

      1. distribute-lft-in 48.7%

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

      2. metadata-eval 48.7%

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

      3. associate-*r/ 48.7%

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

      4. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

      1. flip-- 48.7%

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

      2. metadata-eval 48.7%

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

      3. add-sqr-sqrt 48.8%

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

      4. associate--r+ 48.8%

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

      5. metadata-eval 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in x around 0 98.1%

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

      1. *-commutative 98.1%

         \[\leadsto \color{blue}{{x}^{2} \cdot 0.125} \]

      2. unpow2 98.1%

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

      3. associate-*l* 98.1%

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

   8. Simplified 98.1%

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

4. Final simplification 62.8%

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

Alternative 12: 51.9% 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 72.0%

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

   1. distribute-lft-in 72.0%

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

   2. metadata-eval 72.0%

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

   3. associate-*r/ 72.0%

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

   4. metadata-eval 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in x around 0 54.1%

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

   1. unpow2 54.1%

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

6. Simplified 54.1%

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

7. Final simplification 54.1%

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

Alternative 13: 51.9% accurate, 42.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. distribute-lft-in 72.0%

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

   2. metadata-eval 72.0%

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

   3. associate-*r/ 72.0%

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

   4. metadata-eval 72.0%

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

3. Simplified 72.0%

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

   1. flip-- 72.0%

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

   2. metadata-eval 72.0%

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

   3. add-sqr-sqrt 72.8%

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

   4. associate--r+ 72.8%

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

   5. metadata-eval 72.8%

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

5. Applied egg-rr 72.8%

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

6. Taylor expanded in x around 0 54.1%

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

   1. *-commutative 54.1%

      \[\leadsto \color{blue}{{x}^{2} \cdot 0.125} \]

   2. unpow2 54.1%

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

   3. associate-*l* 54.1%

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

8. Simplified 54.1%

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

9. Final simplification 54.1%

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

Alternative 14: 27.7% 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 72.0%

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

   1. distribute-lft-in 72.0%

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

   2. metadata-eval 72.0%

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

   3. associate-*r/ 72.0%

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

   4. metadata-eval 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in x around 0 25.9%

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

5. Final simplification 25.9%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))