Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 99.8%

Time: 13.6s

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: 76.3% 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

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (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) 0.25) 2.0))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = 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), 0.25), 2.0));
	}
	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) <= 2.0) {
		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), 0.25), 2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		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), 0.25), 2.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = 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) ^ 0.25) ^ 2.0)));
	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) <= 2.0)
		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) ^ 0.25) ^ 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]}, 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] * 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], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. distribute-lft-in 52.8%

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

      2. metadata-eval 52.8%

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

      3. associate-*r/ 52.8%

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

      4. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. pow1/2 98.5%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + {\left({\left(0.5 + t\_0\right)}^{0.25}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.00005)
     (* (pow x 2.0) (+ 0.125 (* (pow x 2.0) -0.0859375)))
     (/ (- 0.5 t_0) (+ 1.0 (pow (pow (+ 0.5 t_0) 0.25) 2.0))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.00005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - t_0) / (1.0 + pow(pow((0.5 + t_0), 0.25), 2.0));
	}
	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) <= 1.00005) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * -0.0859375));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.pow(Math.pow((0.5 + t_0), 0.25), 2.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.00005:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * -0.0859375))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.pow(math.pow((0.5 + t_0), 0.25), 2.0))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.00005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * -0.0859375)));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + ((Float64(0.5 + t_0) ^ 0.25) ^ 2.0)));
	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) <= 1.00005)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * -0.0859375));
	else
		tmp = (0.5 - t_0) / (1.0 + (((0.5 + t_0) ^ 0.25) ^ 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]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.00005], 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 - t$95$0), $MachinePrecision] / N[(1.0 + N[Power[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00005000000000011

   1. Initial program 52.6%

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

      1. distribute-lft-in 52.6%

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

      2. metadata-eval 52.6%

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

      3. associate-*r/ 52.6%

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

      4. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 1.00005000000000011 < (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-sqr-sqrt 98.4%

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

      2. pow2 98.4%

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

      3. pow1/2 98.4%

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

      4. sqrt-pow1 99.8%

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

      5. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{0.25}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.00005)
     (* (pow x 2.0) (+ 0.125 (* (pow x 2.0) -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) <= 1.00005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * -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) <= 1.00005) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * -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) <= 1.00005:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * -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) <= 1.00005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * -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) <= 1.00005)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * -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], 1.00005], 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 - 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) < 1.00005000000000011

   1. Initial program 52.6%

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

      1. distribute-lft-in 52.6%

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

      2. metadata-eval 52.6%

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

      3. associate-*r/ 52.6%

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

      4. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 1.00005000000000011 < (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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot \left({x}^{2} \cdot 0.0673828125 - 0.0859375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + {\left({\left(0.5 + \frac{0.5}{x}\right)}^{0.25}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (pow x 2.0)
    (+ 0.125 (* (pow x 2.0) (- (* (pow x 2.0) 0.0673828125) 0.0859375))))
   (/ (- 0.5 (/ 0.5 x)) (+ 1.0 (pow (pow (+ 0.5 (/ 0.5 x)) 0.25) 2.0)))))

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) * ((pow(x, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + pow(pow((0.5 + (0.5 / x)), 0.25), 2.0));
	}
	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) * ((Math.pow(x, 2.0) * 0.0673828125) - 0.0859375)));
	} else {
		tmp = (0.5 - (0.5 / x)) / (1.0 + Math.pow(Math.pow((0.5 + (0.5 / x)), 0.25), 2.0));
	}
	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) * ((math.pow(x, 2.0) * 0.0673828125) - 0.0859375)))
	else:
		tmp = (0.5 - (0.5 / x)) / (1.0 + math.pow(math.pow((0.5 + (0.5 / x)), 0.25), 2.0))
	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) * Float64(Float64((x ^ 2.0) * 0.0673828125) - 0.0859375))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 / x)) / Float64(1.0 + ((Float64(0.5 + Float64(0.5 / x)) ^ 0.25) ^ 2.0)));
	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) * (((x ^ 2.0) * 0.0673828125) - 0.0859375)));
	else
		tmp = (0.5 - (0.5 / x)) / (1.0 + (((0.5 + (0.5 / x)) ^ 0.25) ^ 2.0));
	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] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.0673828125), $MachinePrecision] - 0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Power[N[Power[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. distribute-lft-in 52.8%

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

      2. metadata-eval 52.8%

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

      3. associate-*r/ 52.8%

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

      4. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in x around 0 99.8%

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

   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. 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. Step-by-step derivation

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. pow1/2 98.5%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 98.4%

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

       1. associate-*r/ 96.8%

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

       2. metadata-eval 96.8%

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

   11. Simplified 98.4%

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

   12. Taylor expanded in x around inf 98.2%

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

       1. associate-*r/ 98.2%

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

       2. metadata-eval 98.2%

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

   14. Simplified 98.2%

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

4. Final simplification 99.0%

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

Alternative 5: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{x}}{1 + {\left({\left(0.5 + \frac{0.5}{x}\right)}^{0.25}\right)}^{2}}\\ \end{array} \end{array} \]

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 (pow (pow (+ 0.5 (/ 0.5 x)) 0.25) 2.0)))))

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 + pow(pow((0.5 + (0.5 / x)), 0.25), 2.0));
	}
	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.pow(Math.pow((0.5 + (0.5 / x)), 0.25), 2.0));
	}
	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.pow(math.pow((0.5 + (0.5 / x)), 0.25), 2.0))
	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 + ((Float64(0.5 + Float64(0.5 / x)) ^ 0.25) ^ 2.0)));
	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 + (((0.5 + (0.5 / x)) ^ 0.25) ^ 2.0));
	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[Power[N[Power[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. distribute-lft-in 52.8%

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

      2. metadata-eval 52.8%

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

      3. associate-*r/ 52.8%

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

      4. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

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

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. add-sqr-sqrt 98.5%

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

      2. pow2 98.5%

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

      3. pow1/2 98.5%

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

      4. sqrt-pow1 100.0%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 98.4%

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

       1. associate-*r/ 96.8%

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

       2. metadata-eval 96.8%

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

   11. Simplified 98.4%

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

   12. Taylor expanded in x around inf 98.2%

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

       1. associate-*r/ 98.2%

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

       2. metadata-eval 98.2%

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

   14. Simplified 98.2%

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

4. Final simplification 98.9%

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

Alternative 6: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.00005)
   (* (pow x 2.0) (+ 0.125 (* (pow x 2.0) -0.0859375)))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.00005) {
		tmp = pow(x, 2.0) * (0.125 + (pow(x, 2.0) * -0.0859375));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.00005) {
		tmp = Math.pow(x, 2.0) * (0.125 + (Math.pow(x, 2.0) * -0.0859375));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.00005:
		tmp = math.pow(x, 2.0) * (0.125 + (math.pow(x, 2.0) * -0.0859375))
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.00005)
		tmp = Float64((x ^ 2.0) * Float64(0.125 + Float64((x ^ 2.0) * -0.0859375)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.00005)
		tmp = (x ^ 2.0) * (0.125 + ((x ^ 2.0) * -0.0859375));
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.00005], N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.125 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00005000000000011

   1. Initial program 52.6%

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

      1. distribute-lft-in 52.6%

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

      2. metadata-eval 52.6%

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

      3. associate-*r/ 52.6%

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

      4. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if 1.00005000000000011 < (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
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;{x}^{2} \cdot \left(0.125 + {x}^{2} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.00005)
   (* 0.125 (* x x))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.00005) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.00005) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.00005:
		tmp = 0.125 * (x * x)
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.00005)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.00005)
		tmp = 0.125 * (x * x);
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.00005], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00005000000000011

   1. Initial program 52.6%

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

      1. distribute-lft-in 52.6%

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

      2. metadata-eval 52.6%

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

      3. associate-*r/ 52.6%

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

      4. metadata-eval 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in x around 0 52.7%

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

      1. *-commutative 52.7%

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

   7. Simplified 52.7%

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

      1. associate--r+ 99.8%

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

      2. metadata-eval 99.8%

         \[\leadsto \color{blue}{0} - {x}^{2} \cdot -0.125 \]

      3. flip-- 33.8%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125}} \]

      4. metadata-eval 33.8%

         \[\leadsto \frac{\color{blue}{0} - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      5. swap-sqr 33.8%

         \[\leadsto \frac{0 - \color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(-0.125 \cdot -0.125\right)}}{0 + {x}^{2} \cdot -0.125} \]

      6. pow-prod-up 33.6%

         \[\leadsto \frac{0 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      7. metadata-eval 33.6%

         \[\leadsto \frac{0 - {x}^{\color{blue}{4}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      8. metadata-eval 33.6%

         \[\leadsto \frac{0 - {x}^{4} \cdot \color{blue}{0.015625}}{0 + {x}^{2} \cdot -0.125} \]

   9. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{0 - {x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]
   10. Step-by-step derivation

       1. div-sub 33.6%

          \[\leadsto \color{blue}{\frac{0}{0 + {x}^{2} \cdot -0.125} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       2. +-lft-identity 33.6%

          \[\leadsto \frac{0}{\color{blue}{{x}^{2} \cdot -0.125}} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       3. div0 33.6%

          \[\leadsto \color{blue}{0} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       4. neg-sub0 33.6%

          \[\leadsto \color{blue}{-\frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       5. *-commutative 33.6%

          \[\leadsto -\frac{\color{blue}{0.015625 \cdot {x}^{4}}}{0 + {x}^{2} \cdot -0.125} \]

       6. +-lft-identity 33.6%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{{x}^{2} \cdot -0.125}} \]

       7. *-commutative 33.6%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{-0.125 \cdot {x}^{2}}} \]

       8. times-frac 33.7%

          \[\leadsto -\color{blue}{\frac{0.015625}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}}} \]

       9. metadata-eval 33.7%

          \[\leadsto -\color{blue}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

       10. distribute-lft-neg-in 33.7%

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

       11. metadata-eval 33.7%

           \[\leadsto \color{blue}{0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

   11. Simplified 33.7%

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

       1. pow-div 99.8%

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

       2. metadata-eval 99.8%

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

       3. unpow2 99.8%

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

   13. Applied egg-rr 99.8%

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

   if 1.00005000000000011 < (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
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00005:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = 0.125 * (x * x);
	else
		tmp = 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[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. distribute-lft-in 52.8%

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

      2. metadata-eval 52.8%

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

      3. associate-*r/ 52.8%

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

      4. metadata-eval 52.8%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in x around 0 52.6%

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

      1. *-commutative 52.6%

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

   7. Simplified 52.6%

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

      1. associate--r+ 99.3%

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

      2. metadata-eval 99.3%

         \[\leadsto \color{blue}{0} - {x}^{2} \cdot -0.125 \]

      3. flip-- 33.8%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125}} \]

      4. metadata-eval 33.8%

         \[\leadsto \frac{\color{blue}{0} - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      5. swap-sqr 33.8%

         \[\leadsto \frac{0 - \color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(-0.125 \cdot -0.125\right)}}{0 + {x}^{2} \cdot -0.125} \]

      6. pow-prod-up 33.6%

         \[\leadsto \frac{0 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      7. metadata-eval 33.6%

         \[\leadsto \frac{0 - {x}^{\color{blue}{4}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      8. metadata-eval 33.6%

         \[\leadsto \frac{0 - {x}^{4} \cdot \color{blue}{0.015625}}{0 + {x}^{2} \cdot -0.125} \]

   9. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{0 - {x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]
   10. Step-by-step derivation

       1. div-sub 33.6%

          \[\leadsto \color{blue}{\frac{0}{0 + {x}^{2} \cdot -0.125} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       2. +-lft-identity 33.6%

          \[\leadsto \frac{0}{\color{blue}{{x}^{2} \cdot -0.125}} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       3. div0 33.6%

          \[\leadsto \color{blue}{0} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       4. neg-sub0 33.6%

          \[\leadsto \color{blue}{-\frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       5. *-commutative 33.6%

          \[\leadsto -\frac{\color{blue}{0.015625 \cdot {x}^{4}}}{0 + {x}^{2} \cdot -0.125} \]

       6. +-lft-identity 33.6%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{{x}^{2} \cdot -0.125}} \]

       7. *-commutative 33.6%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{-0.125 \cdot {x}^{2}}} \]

       8. times-frac 33.7%

          \[\leadsto -\color{blue}{\frac{0.015625}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}}} \]

       9. metadata-eval 33.7%

          \[\leadsto -\color{blue}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

       10. distribute-lft-neg-in 33.7%

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

       11. metadata-eval 33.7%

           \[\leadsto \color{blue}{0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

   11. Simplified 33.7%

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

       1. pow-div 99.3%

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

       2. metadata-eval 99.3%

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

       3. unpow2 99.3%

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

   13. Applied egg-rr 99.3%

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

   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 96.8%

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

      1. associate-*r/ 96.8%

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

      2. metadata-eval 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 98.0%

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

Alternative 9: 74.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (* 0.125 (* x x)) (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 0.5d0 / (1.0d0 + sqrt(0.5d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = 0.125 * (x * x)
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = 0.125 * (x * x);
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 68.5%

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

      1. distribute-lft-in 68.5%

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

      2. metadata-eval 68.5%

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

      3. associate-*r/ 68.5%

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

      4. metadata-eval 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

   7. Simplified 36.3%

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

      1. associate--r+ 67.1%

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

      2. metadata-eval 67.1%

         \[\leadsto \color{blue}{0} - {x}^{2} \cdot -0.125 \]

      3. flip-- 23.5%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125}} \]

      4. metadata-eval 23.5%

         \[\leadsto \frac{\color{blue}{0} - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      5. swap-sqr 23.5%

         \[\leadsto \frac{0 - \color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(-0.125 \cdot -0.125\right)}}{0 + {x}^{2} \cdot -0.125} \]

      6. pow-prod-up 23.4%

         \[\leadsto \frac{0 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      7. metadata-eval 23.4%

         \[\leadsto \frac{0 - {x}^{\color{blue}{4}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      8. metadata-eval 23.4%

         \[\leadsto \frac{0 - {x}^{4} \cdot \color{blue}{0.015625}}{0 + {x}^{2} \cdot -0.125} \]

   9. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{0 - {x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]
   10. Step-by-step derivation

       1. div-sub 23.4%

          \[\leadsto \color{blue}{\frac{0}{0 + {x}^{2} \cdot -0.125} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       2. +-lft-identity 23.4%

          \[\leadsto \frac{0}{\color{blue}{{x}^{2} \cdot -0.125}} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       3. div0 23.4%

          \[\leadsto \color{blue}{0} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       4. neg-sub0 23.4%

          \[\leadsto \color{blue}{-\frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       5. *-commutative 23.4%

          \[\leadsto -\frac{\color{blue}{0.015625 \cdot {x}^{4}}}{0 + {x}^{2} \cdot -0.125} \]

       6. +-lft-identity 23.4%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{{x}^{2} \cdot -0.125}} \]

       7. *-commutative 23.4%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{-0.125 \cdot {x}^{2}}} \]

       8. times-frac 23.4%

          \[\leadsto -\color{blue}{\frac{0.015625}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}}} \]

       9. metadata-eval 23.4%

          \[\leadsto -\color{blue}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

       10. distribute-lft-neg-in 23.4%

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

       11. metadata-eval 23.4%

           \[\leadsto \color{blue}{0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

   11. Simplified 23.4%

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

       1. pow-div 67.1%

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

       2. metadata-eval 67.1%

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

       3. unpow2 67.1%

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

   13. Applied egg-rr 67.1%

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

   if 1.55000000000000004 < 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 99.8%

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

      1. associate-+r+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 99.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.3% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (* 0.125 (* x x)) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * (x * x);
	} 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 <= 1.55d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = 0.125 * (x * x)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 68.5%

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

      1. distribute-lft-in 68.5%

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

      2. metadata-eval 68.5%

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

      3. associate-*r/ 68.5%

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

      4. metadata-eval 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

   7. Simplified 36.3%

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

      1. associate--r+ 67.1%

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

      2. metadata-eval 67.1%

         \[\leadsto \color{blue}{0} - {x}^{2} \cdot -0.125 \]

      3. flip-- 23.5%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125}} \]

      4. metadata-eval 23.5%

         \[\leadsto \frac{\color{blue}{0} - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      5. swap-sqr 23.5%

         \[\leadsto \frac{0 - \color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(-0.125 \cdot -0.125\right)}}{0 + {x}^{2} \cdot -0.125} \]

      6. pow-prod-up 23.4%

         \[\leadsto \frac{0 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      7. metadata-eval 23.4%

         \[\leadsto \frac{0 - {x}^{\color{blue}{4}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      8. metadata-eval 23.4%

         \[\leadsto \frac{0 - {x}^{4} \cdot \color{blue}{0.015625}}{0 + {x}^{2} \cdot -0.125} \]

   9. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{0 - {x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]
   10. Step-by-step derivation

       1. div-sub 23.4%

          \[\leadsto \color{blue}{\frac{0}{0 + {x}^{2} \cdot -0.125} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       2. +-lft-identity 23.4%

          \[\leadsto \frac{0}{\color{blue}{{x}^{2} \cdot -0.125}} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       3. div0 23.4%

          \[\leadsto \color{blue}{0} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       4. neg-sub0 23.4%

          \[\leadsto \color{blue}{-\frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       5. *-commutative 23.4%

          \[\leadsto -\frac{\color{blue}{0.015625 \cdot {x}^{4}}}{0 + {x}^{2} \cdot -0.125} \]

       6. +-lft-identity 23.4%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{{x}^{2} \cdot -0.125}} \]

       7. *-commutative 23.4%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{-0.125 \cdot {x}^{2}}} \]

       8. times-frac 23.4%

          \[\leadsto -\color{blue}{\frac{0.015625}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}}} \]

       9. metadata-eval 23.4%

          \[\leadsto -\color{blue}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

       10. distribute-lft-neg-in 23.4%

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

       11. metadata-eval 23.4%

           \[\leadsto \color{blue}{0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

   11. Simplified 23.4%

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

       1. pow-div 67.1%

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

       2. metadata-eval 67.1%

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

       3. unpow2 67.1%

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

   13. Applied egg-rr 67.1%

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

   if 1.55000000000000004 < 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 97.6%

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

4. Final simplification 74.6%

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

Alternative 11: 56.3% accurate, 21.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = 0.125 * (x * x);
	} 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 = 0.125d0 * (x * x)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 68.5%

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

      1. distribute-lft-in 68.5%

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

      2. metadata-eval 68.5%

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

      3. associate-*r/ 68.5%

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

      4. metadata-eval 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in x around 0 36.3%

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

      1. *-commutative 36.3%

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

   7. Simplified 36.3%

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

      1. associate--r+ 67.1%

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

      2. metadata-eval 67.1%

         \[\leadsto \color{blue}{0} - {x}^{2} \cdot -0.125 \]

      3. flip-- 23.5%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125}} \]

      4. metadata-eval 23.5%

         \[\leadsto \frac{\color{blue}{0} - \left({x}^{2} \cdot -0.125\right) \cdot \left({x}^{2} \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      5. swap-sqr 23.5%

         \[\leadsto \frac{0 - \color{blue}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(-0.125 \cdot -0.125\right)}}{0 + {x}^{2} \cdot -0.125} \]

      6. pow-prod-up 23.4%

         \[\leadsto \frac{0 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      7. metadata-eval 23.4%

         \[\leadsto \frac{0 - {x}^{\color{blue}{4}} \cdot \left(-0.125 \cdot -0.125\right)}{0 + {x}^{2} \cdot -0.125} \]

      8. metadata-eval 23.4%

         \[\leadsto \frac{0 - {x}^{4} \cdot \color{blue}{0.015625}}{0 + {x}^{2} \cdot -0.125} \]

   9. Applied egg-rr 23.4%

      \[\leadsto \color{blue}{\frac{0 - {x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]
   10. Step-by-step derivation

       1. div-sub 23.4%

          \[\leadsto \color{blue}{\frac{0}{0 + {x}^{2} \cdot -0.125} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       2. +-lft-identity 23.4%

          \[\leadsto \frac{0}{\color{blue}{{x}^{2} \cdot -0.125}} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       3. div0 23.4%

          \[\leadsto \color{blue}{0} - \frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125} \]

       4. neg-sub0 23.4%

          \[\leadsto \color{blue}{-\frac{{x}^{4} \cdot 0.015625}{0 + {x}^{2} \cdot -0.125}} \]

       5. *-commutative 23.4%

          \[\leadsto -\frac{\color{blue}{0.015625 \cdot {x}^{4}}}{0 + {x}^{2} \cdot -0.125} \]

       6. +-lft-identity 23.4%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{{x}^{2} \cdot -0.125}} \]

       7. *-commutative 23.4%

          \[\leadsto -\frac{0.015625 \cdot {x}^{4}}{\color{blue}{-0.125 \cdot {x}^{2}}} \]

       8. times-frac 23.4%

          \[\leadsto -\color{blue}{\frac{0.015625}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}}} \]

       9. metadata-eval 23.4%

          \[\leadsto -\color{blue}{-0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

       10. distribute-lft-neg-in 23.4%

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

       11. metadata-eval 23.4%

           \[\leadsto \color{blue}{0.125} \cdot \frac{{x}^{4}}{{x}^{2}} \]

   11. Simplified 23.4%

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

       1. pow-div 67.1%

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

       2. metadata-eval 67.1%

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

       3. unpow2 67.1%

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

   13. Applied egg-rr 67.1%

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

   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%

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

   8. Taylor expanded in x around inf 22.7%

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

4. Final simplification 56.1%

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

Alternative 12: 33.0% 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 76.2%

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

      1. distribute-lft-in 76.2%

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

      2. metadata-eval 76.2%

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

      3. associate-*r/ 76.2%

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

      4. metadata-eval 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in x around 0 39.0%

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

   if 2.10000000000000015e-77 < x

   1. Initial program 75.1%

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

      1. distribute-lft-in 75.1%

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

      2. metadata-eval 75.1%

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

      3. associate-*r/ 75.1%

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

      4. metadata-eval 75.1%

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

   3. Simplified 75.1%

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

      1. flip-- 75.1%

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

      2. metadata-eval 75.1%

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

      3. add-sqr-sqrt 76.3%

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

      4. associate--r+ 76.3%

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

      5. metadata-eval 76.3%

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

   6. Applied egg-rr 76.3%

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

   7. Taylor expanded in x around 0 18.4%

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

   8. Taylor expanded in x around inf 18.8%

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

4. Final simplification 32.4%

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

Alternative 13: 28.1% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 75.8%

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

   1. distribute-lft-in 75.8%

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

   2. metadata-eval 75.8%

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

   3. associate-*r/ 75.8%

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

   4. metadata-eval 75.8%

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

3. Simplified 75.8%

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

5. Taylor expanded in x around 0 27.3%

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

6. Final simplification 27.3%

   \[\leadsto 0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))