Given's Rotation SVD example, simplified

Percentage Accurate: 76.4% → 99.8%

Time: 16.5s

Alternatives: 13

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: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + {\left({\left(0.5 + t\_0\right)}^{0.16666666666666666}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.2)
     (*
      (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 (pow (pow (+ 0.5 t_0) 0.16666666666666666) 3.0))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		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 + pow(pow((0.5 + t_0), 0.16666666666666666), 3.0));
	}
	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) <= 1.2) {
		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.pow(Math.pow((0.5 + t_0), 0.16666666666666666), 3.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.2:
		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.pow(math.pow((0.5 + t_0), 0.16666666666666666), 3.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		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 + ((Float64(0.5 + t_0) ^ 0.16666666666666666) ^ 3.0)));
	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) <= 1.2)
		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 + (((0.5 + t_0) ^ 0.16666666666666666) ^ 3.0));
	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], 1.2], 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[Power[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.19999999999999996

   1. Initial program 49.3%

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

      1. distribute-lft-in 49.3%

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

      2. metadata-eval 49.3%

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

      3. associate-*r/ 49.3%

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

      4. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in x around 0 99.9%

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

   if 1.19999999999999996 < (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. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      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. Step-by-step derivation

      1. add-cube-cbrt 98.5%

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

      2. associate-*l* 98.5%

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

      3. pow1/3 98.5%

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

      4. pow1/2 98.5%

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

      5. pow-pow 98.5%

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

      6. metadata-eval 98.5%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)} \]

      7. cbrt-prod 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}} \]

      8. add-sqr-sqrt 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
   9. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.3333333333333333}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}} \]

      3. pow-sqr 100.0%

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

      4. cube-mult 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666}\right)}^{3}}} \]

   10. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.0673828125 + {x}^{2} \cdot -0.056243896484375\right) - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + {\left({\left(0.5 + t\_0\right)}^{0.16666666666666666}\right)}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.2)
     (*
      (pow x 2.0)
      (+ 0.125 (* (pow x 2.0) (- (* (pow x 2.0) 0.0673828125) 0.0859375))))
     (/ (- 0.5 t_0) (+ 1.0 (pow (pow (+ 0.5 t_0) 0.16666666666666666) 3.0))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + pow(pow((0.5 + t_0), 0.16666666666666666), 3.0));
	}
	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) <= 1.2) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.pow(Math.pow((0.5 + t_0), 0.16666666666666666), 3.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.2:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.0673828125) - 0.0859375)))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.pow(math.pow((0.5 + t_0), 0.16666666666666666), 3.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.0673828125) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + ((Float64(0.5 + t_0) ^ 0.16666666666666666) ^ 3.0)));
	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) <= 1.2)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * (((x ^ 2.0) * 0.0673828125) - 0.0859375)));
	else
		tmp = (0.5 - t_0) / (1.0 + (((0.5 + t_0) ^ 0.16666666666666666) ^ 3.0));
	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], 1.2], 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] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Power[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 0.16666666666666666], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.19999999999999996

   1. Initial program 49.3%

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

      1. distribute-lft-in 49.3%

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

      2. metadata-eval 49.3%

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

      3. associate-*r/ 49.3%

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

      4. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in x around 0 99.9%

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

   if 1.19999999999999996 < (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. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      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. Step-by-step derivation

      1. add-cube-cbrt 98.5%

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

      2. associate-*l* 98.5%

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

      3. pow1/3 98.5%

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

      4. pow1/2 98.5%

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

      5. pow-pow 98.5%

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

      6. metadata-eval 98.5%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\right)} \]

      7. cbrt-prod 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}} \]

      8. add-sqr-sqrt 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

   8. Applied egg-rr 100.0%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \sqrt[3]{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
   9. Step-by-step derivation

      1. unpow1/3 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.3333333333333333}}} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666} \cdot {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}} \]

      3. pow-sqr 100.0%

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

      4. cube-mult 100.0%

         \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \color{blue}{{\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666}\right)}^{3}}} \]

   10. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.16666666666666666}\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.2)
     (*
      (pow x 2.0)
      (+ 0.125 (* (pow x 2.0) (- (* (pow x 2.0) 0.0673828125) 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) <= 1.2) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.0673828125) - 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) <= 1.2) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.0673828125) - 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) <= 1.2:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.0673828125) - 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) <= 1.2)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.0673828125) - 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) <= 1.2)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * (((x ^ 2.0) * 0.0673828125) - 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], 1.2], 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] * 0.0673828125), $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 1 x) < 1.19999999999999996

   1. Initial program 49.3%

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

      1. distribute-lft-in 49.3%

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

      2. metadata-eval 49.3%

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

      3. associate-*r/ 49.3%

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

      4. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in x around 0 99.9%

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

   if 1.19999999999999996 < (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. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.2)
   (*
    (pow x 2.0)
    (+ 0.125 (* (pow x 2.0) (- (* (pow x 2.0) 0.0673828125) 0.0859375))))
   (/ (- 0.5 (/ 0.5 (hypot 1.0 x))) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / hypot(1.0, 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) <= 1.2) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.2:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.0673828125) - 0.0859375)))
	else:
		tmp = (0.5 - (0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.0673828125) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / hypot(1.0, 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) <= 1.2)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * (((x ^ 2.0) * 0.0673828125) - 0.0859375)));
	else
		tmp = (0.5 - (0.5 / hypot(1.0, 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], 1.2], 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] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.19999999999999996

   1. Initial program 49.3%

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

      1. distribute-lft-in 49.3%

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

      2. metadata-eval 49.3%

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

      3. associate-*r/ 49.3%

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

      4. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in x around 0 99.9%

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

   if 1.19999999999999996 < (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. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      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%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.2)
   (* (pow x 2.0) (+ 0.125 (* (pow x 2.0) -0.0859375)))
   (/ (- 0.5 (/ 0.5 (hypot 1.0 x))) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - (0.5 / hypot(1.0, 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) <= 1.2) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - (0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.2:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * -0.0859375))
	else:
		tmp = (0.5 - (0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * -0.0859375)));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / hypot(1.0, 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) <= 1.2)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * -0.0859375));
	else
		tmp = (0.5 - (0.5 / hypot(1.0, 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], 1.2], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.19999999999999996

   1. Initial program 49.3%

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

      1. distribute-lft-in 49.3%

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

      2. metadata-eval 49.3%

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

      3. associate-*r/ 49.3%

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

      4. metadata-eval 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   if 1.19999999999999996 < (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. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      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%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (pow x 2.0) (+ 0.125 (* (pow x 2.0) -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 + (pow(x, 2.0) * -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 + (Math.pow(x, 2.0) * -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 + (math.pow(x, 2.0) * -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((x ^ 2.0) * -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 ^ 2.0) * -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[Power[x, 2.0], $MachinePrecision] * -0.0859375), $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 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around inf 97.9%

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

      1. flip-- 97.9%

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

      2. metadata-eval 97.9%

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

      3. add-sqr-sqrt 99.4%

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

      4. associate--r+ 99.4%

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

      5. metadata-eval 99.4%

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 99.2%

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

Alternative 7: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = x * (x * 0.125)
	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 * Float64(x * 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 * (x * 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[(x * N[(x * 0.125), $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 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around 0 48.8%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 48.8%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]
   8. Step-by-step derivation

      1. +-commutative 48.8%

         \[\leadsto 1 - \color{blue}{\left({x}^{2} \cdot -0.125 + 1\right)} \]

      2. associate--r+ 48.7%

         \[\leadsto \color{blue}{\left(1 - {x}^{2} \cdot -0.125\right) - 1} \]

      3. add-exp-log 48.7%

         \[\leadsto \color{blue}{e^{\log \left(1 - {x}^{2} \cdot -0.125\right)}} - 1 \]

      4. *-commutative 48.7%

         \[\leadsto e^{\log \left(1 - \color{blue}{-0.125 \cdot {x}^{2}}\right)} - 1 \]

      5. cancel-sign-sub-inv 48.7%

         \[\leadsto e^{\log \color{blue}{\left(1 + \left(--0.125\right) \cdot {x}^{2}\right)}} - 1 \]

      6. metadata-eval 48.7%

         \[\leadsto e^{\log \left(1 + \color{blue}{0.125} \cdot {x}^{2}\right)} - 1 \]

      7. log1p-undefine 48.7%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      8. expm1-undefine 98.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      9. expm1-log1p-u 98.0%

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

      10. unpow2 98.0%

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

      11. associate-*r* 98.0%

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

   9. Applied egg-rr 98.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around inf 97.9%

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

      1. flip-- 97.9%

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

      2. metadata-eval 97.9%

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

      3. add-sqr-sqrt 99.4%

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

      4. associate--r+ 99.4%

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

      5. metadata-eval 99.4%

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

   7. Applied egg-rr 99.4%

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

4. Final simplification 98.7%

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

Alternative 8: 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 49.7%

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

      1. distribute-lft-in 49.7%

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

      2. metadata-eval 49.7%

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

      3. associate-*r/ 49.7%

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

      4. metadata-eval 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in x around 0 48.8%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 48.8%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 48.8%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]
   8. Step-by-step derivation

      1. +-commutative 48.8%

         \[\leadsto 1 - \color{blue}{\left({x}^{2} \cdot -0.125 + 1\right)} \]

      2. associate--r+ 48.7%

         \[\leadsto \color{blue}{\left(1 - {x}^{2} \cdot -0.125\right) - 1} \]

      3. add-exp-log 48.7%

         \[\leadsto \color{blue}{e^{\log \left(1 - {x}^{2} \cdot -0.125\right)}} - 1 \]

      4. *-commutative 48.7%

         \[\leadsto e^{\log \left(1 - \color{blue}{-0.125 \cdot {x}^{2}}\right)} - 1 \]

      5. cancel-sign-sub-inv 48.7%

         \[\leadsto e^{\log \color{blue}{\left(1 + \left(--0.125\right) \cdot {x}^{2}\right)}} - 1 \]

      6. metadata-eval 48.7%

         \[\leadsto e^{\log \left(1 + \color{blue}{0.125} \cdot {x}^{2}\right)} - 1 \]

      7. log1p-undefine 48.7%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      8. expm1-undefine 98.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      9. expm1-log1p-u 98.0%

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

      10. unpow2 98.0%

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

      11. associate-*r* 98.0%

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

   9. Applied egg-rr 98.0%

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

   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. 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 98.6%

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

4. Final simplification 98.3%

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

Alternative 9: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.52) (* x (* x 0.125)) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.52) {
		tmp = x * (x * 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 <= 1.52d0) then
        tmp = x * (x * 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 <= 1.52) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.52:
		tmp = x * (x * 0.125)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.52)
		tmp = Float64(x * Float64(x * 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 <= 1.52)
		tmp = x * (x * 0.125);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.52], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 66.1%

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

      1. distribute-lft-in 66.1%

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

      2. metadata-eval 66.1%

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

      3. associate-*r/ 66.1%

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

      4. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in x around 0 33.8%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 33.8%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 33.8%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]
   8. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto 1 - \color{blue}{\left({x}^{2} \cdot -0.125 + 1\right)} \]

      2. associate--r+ 33.8%

         \[\leadsto \color{blue}{\left(1 - {x}^{2} \cdot -0.125\right) - 1} \]

      3. add-exp-log 33.8%

         \[\leadsto \color{blue}{e^{\log \left(1 - {x}^{2} \cdot -0.125\right)}} - 1 \]

      4. *-commutative 33.8%

         \[\leadsto e^{\log \left(1 - \color{blue}{-0.125 \cdot {x}^{2}}\right)} - 1 \]

      5. cancel-sign-sub-inv 33.8%

         \[\leadsto e^{\log \color{blue}{\left(1 + \left(--0.125\right) \cdot {x}^{2}\right)}} - 1 \]

      6. metadata-eval 33.8%

         \[\leadsto e^{\log \left(1 + \color{blue}{0.125} \cdot {x}^{2}\right)} - 1 \]

      7. log1p-undefine 33.8%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      8. expm1-undefine 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      9. expm1-log1p-u 66.4%

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

      10. unpow2 66.4%

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

      11. associate-*r* 66.4%

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

   9. Applied egg-rr 66.4%

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

   if 1.52 < 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 96.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.52:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.7% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 66.1%

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

      1. distribute-lft-in 66.1%

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

      2. metadata-eval 66.1%

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

      3. associate-*r/ 66.1%

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

      4. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in x around 0 33.8%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 33.8%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 33.8%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]
   8. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto 1 - \color{blue}{\left({x}^{2} \cdot -0.125 + 1\right)} \]

      2. associate--r+ 33.8%

         \[\leadsto \color{blue}{\left(1 - {x}^{2} \cdot -0.125\right) - 1} \]

      3. add-exp-log 33.8%

         \[\leadsto \color{blue}{e^{\log \left(1 - {x}^{2} \cdot -0.125\right)}} - 1 \]

      4. *-commutative 33.8%

         \[\leadsto e^{\log \left(1 - \color{blue}{-0.125 \cdot {x}^{2}}\right)} - 1 \]

      5. cancel-sign-sub-inv 33.8%

         \[\leadsto e^{\log \color{blue}{\left(1 + \left(--0.125\right) \cdot {x}^{2}\right)}} - 1 \]

      6. metadata-eval 33.8%

         \[\leadsto e^{\log \left(1 + \color{blue}{0.125} \cdot {x}^{2}\right)} - 1 \]

      7. log1p-undefine 33.8%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      8. expm1-undefine 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      9. expm1-log1p-u 66.4%

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

      10. unpow2 66.4%

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

      11. associate-*r* 66.4%

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

   9. Applied egg-rr 66.4%

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

   if 1.3999999999999999 < 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 0 22.7%

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

   8. Taylor expanded in x around inf 22.7%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75) (* x (* x 0.125)) (+ 0.25 (/ 0.25 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25 + (0.25 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.75d0) then
        tmp = x * (x * 0.125d0)
    else
        tmp = 0.25d0 + (0.25d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25 + (0.25 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = x * (x * 0.125)
	else:
		tmp = 0.25 + (0.25 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = Float64(0.25 + Float64(0.25 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = x * (x * 0.125);
	else
		tmp = 0.25 + (0.25 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(0.25 + N[(0.25 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 66.1%

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

      1. distribute-lft-in 66.1%

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

      2. metadata-eval 66.1%

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

      3. associate-*r/ 66.1%

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

      4. metadata-eval 66.1%

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

   3. Simplified 66.1%

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

   5. Taylor expanded in x around 0 33.8%

      \[\leadsto 1 - \color{blue}{\left(1 + -0.125 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 33.8%

         \[\leadsto 1 - \left(1 + \color{blue}{{x}^{2} \cdot -0.125}\right) \]

   7. Simplified 33.8%

      \[\leadsto 1 - \color{blue}{\left(1 + {x}^{2} \cdot -0.125\right)} \]
   8. Step-by-step derivation

      1. +-commutative 33.8%

         \[\leadsto 1 - \color{blue}{\left({x}^{2} \cdot -0.125 + 1\right)} \]

      2. associate--r+ 33.8%

         \[\leadsto \color{blue}{\left(1 - {x}^{2} \cdot -0.125\right) - 1} \]

      3. add-exp-log 33.8%

         \[\leadsto \color{blue}{e^{\log \left(1 - {x}^{2} \cdot -0.125\right)}} - 1 \]

      4. *-commutative 33.8%

         \[\leadsto e^{\log \left(1 - \color{blue}{-0.125 \cdot {x}^{2}}\right)} - 1 \]

      5. cancel-sign-sub-inv 33.8%

         \[\leadsto e^{\log \color{blue}{\left(1 + \left(--0.125\right) \cdot {x}^{2}\right)}} - 1 \]

      6. metadata-eval 33.8%

         \[\leadsto e^{\log \left(1 + \color{blue}{0.125} \cdot {x}^{2}\right)} - 1 \]

      7. log1p-undefine 33.8%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)}} - 1 \]

      8. expm1-undefine 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2}\right)\right)} \]

      9. expm1-log1p-u 66.4%

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

      10. unpow2 66.4%

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

      11. associate-*r* 66.4%

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

   9. Applied egg-rr 66.4%

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

   if 1.75 < 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 0 22.7%

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

   8. Taylor expanded in x around -inf 22.7%

      \[\leadsto \color{blue}{0.25 + 0.25 \cdot \frac{1}{x}} \]
   9. 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%

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

   10. Simplified 22.7%

       \[\leadsto \color{blue}{0.25 + \frac{0.25}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 + \frac{0.25}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.6% 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 73.9%

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

      1. distribute-lft-in 73.9%

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

      2. metadata-eval 73.9%

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

      3. associate-*r/ 73.9%

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

      4. metadata-eval 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in x around 0 36.1%

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

   if 2.10000000000000015e-77 < x

   1. Initial program 77.4%

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

      1. distribute-lft-in 77.4%

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

      2. metadata-eval 77.4%

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

      3. associate-*r/ 77.4%

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

      4. metadata-eval 77.4%

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

   3. Simplified 77.4%

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

      1. flip-- 77.5%

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

      2. metadata-eval 77.5%

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

      3. add-sqr-sqrt 78.6%

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

      4. associate--r+ 78.6%

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

      5. metadata-eval 78.6%

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

   6. Applied egg-rr 78.6%

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

   7. Taylor expanded in x around 0 19.4%

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

   8. Taylor expanded in x around inf 18.9%

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

4. Final simplification 29.8%

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

Alternative 13: 13.4% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 75.2%

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

   1. distribute-lft-in 75.2%

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

   2. metadata-eval 75.2%

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

   3. associate-*r/ 75.2%

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

   4. metadata-eval 75.2%

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

3. Simplified 75.2%

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

   1. flip-- 75.2%

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

   2. metadata-eval 75.2%

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

   3. add-sqr-sqrt 76.1%

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

   4. associate--r+ 76.1%

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

   5. metadata-eval 76.1%

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

6. Applied egg-rr 76.1%

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

7. Taylor expanded in x around 0 35.2%

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

8. Taylor expanded in x around inf 14.1%

   \[\leadsto \color{blue}{0.25} \]

9. Final simplification 14.1%

   \[\leadsto 0.25 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))