Given's Rotation SVD example, simplified

Percentage Accurate: 75.1% → 100.0%

Time: 9.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% 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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \sqrt{0.5 + t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 + 0.5 \cdot t\_1\right) + t\_0 \cdot \left(-1 - t\_1\right)}{{\left(1 + t\_1\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (sqrt (+ 0.5 t_0))))
   (if (<= (hypot 1.0 x) 1.0005)
     (*
      (pow x 2.0)
      (+
       0.125
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.0673828125 (* (pow x 2.0) -0.056243896484375)))
         0.0859375))))
     (/ (+ (+ 0.5 (* 0.5 t_1)) (* t_0 (- -1.0 t_1))) (pow (+ 1.0 t_1) 2.0)))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = sqrt((0.5 + t_0));
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.0673828125 + (pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = ((0.5 + (0.5 * t_1)) + (t_0 * (-1.0 - t_1))) / pow((1.0 + t_1), 2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = Math.sqrt((0.5 + t_0));
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.0673828125 + (Math.pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = ((0.5 + (0.5 * t_1)) + (t_0 * (-1.0 - t_1))) / Math.pow((1.0 + t_1), 2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = math.sqrt((0.5 + t_0))
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.0673828125 + (math.pow(x, 2.0) * -0.056243896484375))) - 0.0859375)))
	else:
		tmp = ((0.5 + (0.5 * t_1)) + (t_0 * (-1.0 - t_1))) / math.pow((1.0 + t_1), 2.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = sqrt(Float64(0.5 + t_0))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.0673828125 + Float64((x ^ 2.0) * -0.056243896484375))) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.5 * t_1)) + Float64(t_0 * Float64(-1.0 - t_1))) / (Float64(1.0 + t_1) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = sqrt((0.5 + t_0));
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * (((x ^ 2.0) * (0.0673828125 + ((x ^ 2.0) * -0.056243896484375))) - 0.0859375)));
	else
		tmp = ((0.5 + (0.5 * t_1)) + (t_0 * (-1.0 - t_1))) / ((1.0 + t_1) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.0673828125 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.056243896484375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 50.8%

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

      1. distribute-lft-in 50.8%

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

      2. metadata-eval 50.8%

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

      3. associate-*r/ 50.8%

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

      4. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + -0.056243896484375 \cdot {x}^{2}\right) - 0.0859375\right)\right)} \]

   if 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. pow1/2 98.3%

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

      2. add-cube-cbrt 98.3%

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

      3. pow3 98.3%

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

      4. pow-pow 98.3%

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

      5. metadata-eval 98.3%

         \[\leadsto 1 - {\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{\color{blue}{1.5}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{1.5}} \]
   7. Step-by-step derivation

      1. pow1/3 98.3%

         \[\leadsto 1 - {\color{blue}{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.3333333333333333}\right)}}^{1.5} \]

      2. pow-pow 98.3%

         \[\leadsto 1 - \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]

      3. metadata-eval 98.3%

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

      4. pow1/2 98.3%

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

      5. flip-- 98.3%

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

      6. metadata-eval 98.3%

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

      7. add-sqr-sqrt 99.8%

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

      8. associate--r+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. div-sub 99.8%

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

      11. frac-sub 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. distribute-lft-in 99.9%

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

      2. metadata-eval 99.9%

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

      3. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 + 0.5 \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) + \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(-1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\\ t_1 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25 + t\_0}{-0.5 + t\_1}}{1 + \sqrt{\frac{0.25 - t\_0}{0.5 + t\_1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.25 (fma x x 1.0))) (t_1 (/ -0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (*
      (pow x 2.0)
      (+
       0.125
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.0673828125 (* (pow x 2.0) -0.056243896484375)))
         0.0859375))))
     (/
      (/ (+ -0.25 t_0) (+ -0.5 t_1))
      (+ 1.0 (sqrt (/ (- 0.25 t_0) (+ 0.5 t_1))))))))

C

double code(double x) {
	double t_0 = 0.25 / fma(x, x, 1.0);
	double t_1 = -0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.0673828125 + (pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = ((-0.25 + t_0) / (-0.5 + t_1)) / (1.0 + sqrt(((0.25 - t_0) / (0.5 + t_1))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.25 / fma(x, x, 1.0))
	t_1 = Float64(-0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.0673828125 + Float64((x ^ 2.0) * -0.056243896484375))) - 0.0859375))));
	else
		tmp = Float64(Float64(Float64(-0.25 + t_0) / Float64(-0.5 + t_1)) / Float64(1.0 + sqrt(Float64(Float64(0.25 - t_0) / Float64(0.5 + t_1)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.0673828125 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.056243896484375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 + t$95$0), $MachinePrecision] / N[(-0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(0.25 - t$95$0), $MachinePrecision] / N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 50.8%

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

      1. distribute-lft-in 50.8%

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

      2. metadata-eval 50.8%

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

      3. associate-*r/ 50.8%

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

      4. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + -0.056243896484375 \cdot {x}^{2}\right) - 0.0859375\right)\right)} \]

   if 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. pow1/2 98.3%

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

      2. add-cube-cbrt 98.3%

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

      3. pow3 98.3%

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

      4. pow-pow 98.3%

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

      5. metadata-eval 98.3%

         \[\leadsto 1 - {\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{\color{blue}{1.5}} \]

   6. Applied egg-rr 98.3%

      \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{1.5}} \]
   7. Step-by-step derivation

      1. pow1/3 98.3%

         \[\leadsto 1 - {\color{blue}{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.3333333333333333}\right)}}^{1.5} \]

      2. pow-pow 98.3%

         \[\leadsto 1 - \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]

      3. metadata-eval 98.3%

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

      4. pow1/2 98.3%

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

      5. flip-- 98.3%

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

      6. metadata-eval 98.3%

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

      7. add-sqr-sqrt 99.8%

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

      8. associate--r+ 99.8%

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

      9. metadata-eval 99.8%

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

      10. flip-- 99.8%

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

      11. associate-/l/ 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{0.25}{1 + {x}^{2}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \]
   9. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

       1. flip-- 99.8%

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

       2. metadata-eval 99.8%

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

       3. frac-times 99.8%

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

       4. metadata-eval 99.8%

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

       5. hypot-undefine 99.8%

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

       6. hypot-undefine 99.8%

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

       7. rem-square-sqrt 99.9%

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

       8. metadata-eval 99.9%

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

       9. pow2 99.9%

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

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{-0.25 + \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\color{blue}{\frac{0.25 - \frac{0.25}{1 + {x}^{2}}}{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}} \]
   13. Step-by-step derivation

       1. +-commutative 99.9%

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

       2. unpow2 99.9%

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

       3. fma-undefine 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25 + \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{0.3333333333333333}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0005)
   (*
    (pow x 2.0)
    (+
     0.125
     (*
      (pow x 2.0)
      (-
       (* (pow x 2.0) (+ 0.0673828125 (* (pow x 2.0) -0.056243896484375)))
       0.0859375))))
   (/
    (pow (pow (+ 0.5 (/ -0.5 (hypot 1.0 x))) 3.0) 0.3333333333333333)
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.0673828125 + (pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = pow(pow((0.5 + (-0.5 / hypot(1.0, x))), 3.0), 0.3333333333333333) / (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) <= 1.0005) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.0673828125 + (Math.pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = Math.pow(Math.pow((0.5 + (-0.5 / Math.hypot(1.0, x))), 3.0), 0.3333333333333333) / (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) <= 1.0005:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.0673828125 + (math.pow(x, 2.0) * -0.056243896484375))) - 0.0859375)))
	else:
		tmp = math.pow(math.pow((0.5 + (-0.5 / math.hypot(1.0, x))), 3.0), 0.3333333333333333) / (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) <= 1.0005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.0673828125 + Float64((x ^ 2.0) * -0.056243896484375))) - 0.0859375))));
	else
		tmp = Float64(((Float64(0.5 + Float64(-0.5 / hypot(1.0, x))) ^ 3.0) ^ 0.3333333333333333) / 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) <= 1.0005)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * (((x ^ 2.0) * (0.0673828125 + ((x ^ 2.0) * -0.056243896484375))) - 0.0859375)));
	else
		tmp = (((0.5 + (-0.5 / hypot(1.0, x))) ^ 3.0) ^ 0.3333333333333333) / (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], 1.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.0673828125 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.056243896484375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(0.5 + N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $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 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 50.8%

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

      1. distribute-lft-in 50.8%

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

      2. metadata-eval 50.8%

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

      3. associate-*r/ 50.8%

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

      4. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + -0.056243896484375 \cdot {x}^{2}\right) - 0.0859375\right)\right)} \]

   if 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      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%

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

   6. Applied egg-rr 99.8%

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

      1. add-cbrt-cube 98.4%

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

      2. pow1/3 99.9%

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

      3. pow3 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}^{0.3333333333333333}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (*
      (pow x 2.0)
      (+
       0.125
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.0673828125 (* (pow x 2.0) -0.056243896484375)))
         0.0859375))))
     (/ (- 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) <= 1.0005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.0673828125 + (pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.0673828125 + (Math.pow(x, 2.0) * -0.056243896484375))) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.0673828125 + (math.pow(x, 2.0) * -0.056243896484375))) - 0.0859375)))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.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) <= 1.0005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.0673828125 + Float64((x ^ 2.0) * -0.056243896484375))) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * (((x ^ 2.0) * (0.0673828125 + ((x ^ 2.0) * -0.056243896484375))) - 0.0859375)));
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = 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], 1.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.0673828125 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.056243896484375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 50.8%

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

      1. distribute-lft-in 50.8%

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

      2. metadata-eval 50.8%

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

      3. associate-*r/ 50.8%

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

      4. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + -0.056243896484375 \cdot {x}^{2}\right) - 0.0859375\right)\right)} \]

   if 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      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%

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

   6. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + \left(x \cdot x\right) \cdot \left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (*
      (pow x 2.0)
      (+ 0.125 (* (* x x) (- (* 0.0673828125 (* x x)) 0.0859375))))
     (/ (- 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) <= 1.0005) {
		tmp = pow(x, 2.0) * (0.125 + ((x * x) * ((0.0673828125 * (x * x)) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = Math.pow(x, 2.0) * (0.125 + ((x * x) * ((0.0673828125 * (x * x)) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = math.pow(x, 2.0) * (0.125 + ((x * x) * ((0.0673828125 * (x * x)) - 0.0859375)))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.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) <= 1.0005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64(Float64(x * x) * Float64(Float64(0.0673828125 * Float64(x * x)) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (x ^ 2.0) * (0.125 + ((x * x) * ((0.0673828125 * (x * x)) - 0.0859375)));
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = 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], 1.0005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[(x * x), $MachinePrecision] * N[(N[(0.0673828125 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 50.8%

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

      1. distribute-lft-in 50.8%

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

      2. metadata-eval 50.8%

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

      3. associate-*r/ 50.8%

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

      4. metadata-eval 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto {x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot \color{blue}{\left(x \cdot x\right)} - 0.0859375\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 99.9%

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

   9. Applied egg-rr 99.9%

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

   if 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

      1. distribute-lft-in 98.3%

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

      2. metadata-eval 98.3%

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

      3. associate-*r/ 98.3%

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

      4. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. flip-- 98.3%

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

      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%

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

   6. 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. Add Preprocessing

Alternative 6: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.4%

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

      1. distribute-lft-in 51.4%

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

      2. metadata-eval 51.4%

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

      3. associate-*r/ 51.4%

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

      4. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot {x}^{2} - 0.0859375\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.2%

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

   7. Applied egg-rr 99.2%

      \[\leadsto {x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left(0.0673828125 \cdot \color{blue}{\left(x \cdot x\right)} - 0.0859375\right)\right) \]
   8. Step-by-step derivation

      1. unpow2 99.2%

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

   9. Applied egg-rr 99.2%

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

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

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

   11. Simplified 99.9%

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

Alternative 7: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (pow x 2.0) 0.125)
   (/ (- 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 = pow(x, 2.0) * 0.125;
	} 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 = Math.pow(x, 2.0) * 0.125;
	} 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 = math.pow(x, 2.0) * 0.125
	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((x ^ 2.0) * 0.125);
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x ^ 2.0) * 0.125;
	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[Power[x, 2.0], $MachinePrecision] * 0.125), $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 #s(literal 1 binary64) x) < 2

   1. Initial program 51.4%

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

      1. distribute-lft-in 51.4%

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

      2. metadata-eval 51.4%

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

      3. associate-*r/ 51.4%

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

      4. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in x around 0 98.2%

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

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

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. metadata-eval 98.4%

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

   7. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-*r/ 99.9%

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

       2. *-rgt-identity 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 99.1%

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

Alternative 8: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = pow(x, 2.0) * 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 = Math.pow(x, 2.0) * 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 = math.pow(x, 2.0) * 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 ^ 2.0) * 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 ^ 2.0) * 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[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $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 #s(literal 1 binary64) x) < 2

   1. Initial program 51.4%

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

      1. distribute-lft-in 51.4%

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

      2. metadata-eval 51.4%

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

      3. associate-*r/ 51.4%

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

      4. metadata-eval 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in x around 0 98.2%

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

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

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.5%

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

Alternative 9: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5) (* (pow x 2.0) 0.125) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = pow(x, 2.0) * 0.125;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = (x ** 2.0d0) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = Math.pow(x, 2.0) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = math.pow(x, 2.0) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64((x ^ 2.0) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = (x ^ 2.0) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[Power[x, 2.0], $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 67.7%

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

      1. distribute-lft-in 67.7%

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

      2. metadata-eval 67.7%

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

      3. associate-*r/ 67.7%

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

      4. metadata-eval 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around 0 65.6%

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

   if 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. Add Preprocessing

   5. Taylor expanded in x around inf 96.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;{x}^{2} \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.15e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.15e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 73.0%

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

      1. distribute-lft-in 73.0%

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

      2. metadata-eval 73.0%

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

      3. associate-*r/ 73.0%

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

      4. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around 0 35.6%

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

      1. metadata-eval 35.6%

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

   7. Applied egg-rr 35.6%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 79.5%

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

      1. distribute-lft-in 79.5%

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

      2. metadata-eval 79.5%

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

      3. associate-*r/ 79.5%

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

      4. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around inf 78.3%

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

Alternative 11: 31.9% accurate, 34.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000015e-77

   1. Initial program 73.0%

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

      1. distribute-lft-in 73.0%

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

      2. metadata-eval 73.0%

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

      3. associate-*r/ 73.0%

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

      4. metadata-eval 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in x around 0 35.6%

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

      1. metadata-eval 35.6%

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

   7. Applied egg-rr 35.6%

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

   if 2.10000000000000015e-77 < x

   1. Initial program 79.5%

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

      1. distribute-lft-in 79.5%

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

      2. metadata-eval 79.5%

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

      3. associate-*r/ 79.5%

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

      4. metadata-eval 79.5%

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

   3. Simplified 79.5%

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

      1. flip-- 79.5%

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

      2. metadata-eval 79.5%

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

      3. add-sqr-sqrt 80.7%

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

      4. associate--r+ 80.7%

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

      5. metadata-eval 80.7%

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

   6. Applied egg-rr 80.7%

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

   7. Taylor expanded in x around 0 18.9%

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

   8. Taylor expanded in x around inf 19.5%

      \[\leadsto \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 26.9% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 74.9%

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

   1. distribute-lft-in 74.9%

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

   2. metadata-eval 74.9%

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

   3. associate-*r/ 74.9%

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

   4. metadata-eval 74.9%

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

3. Simplified 74.9%

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

5. Taylor expanded in x around 0 26.1%

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

   1. metadata-eval 26.1%

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

7. Applied egg-rr 26.1%

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))