Given's Rotation SVD example, simplified

Percentage Accurate: 76.4% → 99.9%

Time: 22.7s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\left(2 - {t\_1}^{4.5}\right) + -1}{\left(\sqrt{t\_1} + \left(t\_0 + 1.5\right)\right) \cdot \left(1 + \left({t\_1}^{1.5} + {t\_1}^{3}\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+
       (* -0.056243896484375 (pow x 8.0))
       (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0)))))
     (pow
      (pow
       (/
        (+ (- 2.0 (pow t_1 4.5)) -1.0)
        (* (+ (sqrt t_1) (+ t_0 1.5)) (+ 1.0 (+ (pow t_1 1.5) (pow t_1 3.0)))))
       3.0)
      0.3333333333333333))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((-0.056243896484375 * pow(x, 8.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0))));
	} else {
		tmp = pow(pow((((2.0 - pow(t_1, 4.5)) + -1.0) / ((sqrt(t_1) + (t_0 + 1.5)) * (1.0 + (pow(t_1, 1.5) + pow(t_1, 3.0))))), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (Math.hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((-0.056243896484375 * Math.pow(x, 8.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0))));
	} else {
		tmp = Math.pow(Math.pow((((2.0 - Math.pow(t_1, 4.5)) + -1.0) / ((Math.sqrt(t_1) + (t_0 + 1.5)) * (1.0 + (Math.pow(t_1, 1.5) + Math.pow(t_1, 3.0))))), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	tmp = 0
	if math.hypot(1.0, x) <= 1.002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((-0.056243896484375 * math.pow(x, 8.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0))))
	else:
		tmp = math.pow(math.pow((((2.0 - math.pow(t_1, 4.5)) + -1.0) / ((math.sqrt(t_1) + (t_0 + 1.5)) * (1.0 + (math.pow(t_1, 1.5) + math.pow(t_1, 3.0))))), 3.0), 0.3333333333333333)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(-0.056243896484375 * (x ^ 8.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0)))));
	else
		tmp = (Float64(Float64(Float64(2.0 - (t_1 ^ 4.5)) + -1.0) / Float64(Float64(sqrt(t_1) + Float64(t_0 + 1.5)) * Float64(1.0 + Float64((t_1 ^ 1.5) + (t_1 ^ 3.0))))) ^ 3.0) ^ 0.3333333333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = 0.5 + t_0;
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((-0.056243896484375 * (x ^ 8.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0))));
	else
		tmp = ((((2.0 - (t_1 ^ 4.5)) + -1.0) / ((sqrt(t_1) + (t_0 + 1.5)) * (1.0 + ((t_1 ^ 1.5) + (t_1 ^ 3.0))))) ^ 3.0) ^ 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.056243896484375 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(N[(2.0 - N[Power[t$95$1, 4.5], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(N[Sqrt[t$95$1], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[t$95$1, 1.5], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.002

   1. Initial program 57.3%

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

      1. distribute-lft-in 57.3%

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

      2. metadata-eval 57.3%

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

      3. associate-*r/ 57.3%

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

      4. metadata-eval 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)} \]

   if 1.002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip3-- 97.5%

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

      2. div-inv 97.5%

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

      3. metadata-eval 97.5%

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

      4. sqrt-pow2 98.4%

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

      5. metadata-eval 98.4%

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

      6. metadata-eval 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. *-un-lft-identity 98.4%

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

      9. associate-+r+ 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

   8. Simplified 99.9%

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

      1. sub-neg 99.9%

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

      2. flip3-+ 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-un-lft-identity 98.4%

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

   10. Applied egg-rr 98.4%

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

       1. cube-neg 98.4%

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

       2. unsub-neg 98.4%

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

       3. associate-+r- 99.9%

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

       4. sub-neg 99.9%

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

       5. remove-double-neg 99.9%

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

       6. associate-+r+ 98.4%

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

       7. +-commutative 98.4%

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

   12. Simplified 99.9%

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

       1. expm1-log1p-u 98.4%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)}^{3}\right)\right)}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       2. pow-pow 98.4%

          \[\leadsto \frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\left(1.5 \cdot 3\right)}}\right)\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       3. metadata-eval 98.4%

          \[\leadsto \frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{4.5}}\right)\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

   14. Applied egg-rr 98.4%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)\right)}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]
   15. Step-by-step derivation

       1. expm1-undefine 98.4%

          \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)} - 1}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       2. sub-neg 98.4%

          \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)} + \left(-1\right)}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       3. log1p-undefine 99.9%

          \[\leadsto \frac{\frac{e^{\color{blue}{\log \left(1 + \left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)\right)}} + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       4. rem-exp-log 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)\right)} + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       5. associate-+r- 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\left(\left(1 + 1\right) - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)} + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       6. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\left(\color{blue}{2} - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + \color{blue}{-1}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

   16. Simplified 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]
   17. Step-by-step derivation

       1. add-cbrt-cube 98.4%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \cdot \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\right) \cdot \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}} \]

       2. pow1/3 99.9%

          \[\leadsto \color{blue}{{\left(\left(\frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \cdot \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\right) \cdot \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\right)}^{0.3333333333333333}} \]

   18. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{{\left({\left(\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 - {t\_1}^{4.5}\right) + -1}{\left(\sqrt{t\_1} + \left(t\_0 + 1.5\right)\right) \cdot \left(1 + \left({t\_1}^{1.5} + {t\_1}^{3}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+
       (* -0.056243896484375 (pow x 8.0))
       (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0)))))
     (/
      (+ (- 2.0 (pow t_1 4.5)) -1.0)
      (*
       (+ (sqrt t_1) (+ t_0 1.5))
       (+ 1.0 (+ (pow t_1 1.5) (pow t_1 3.0))))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((-0.056243896484375 * pow(x, 8.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0))));
	} else {
		tmp = ((2.0 - pow(t_1, 4.5)) + -1.0) / ((sqrt(t_1) + (t_0 + 1.5)) * (1.0 + (pow(t_1, 1.5) + pow(t_1, 3.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (Math.hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((-0.056243896484375 * Math.pow(x, 8.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0))));
	} else {
		tmp = ((2.0 - Math.pow(t_1, 4.5)) + -1.0) / ((Math.sqrt(t_1) + (t_0 + 1.5)) * (1.0 + (Math.pow(t_1, 1.5) + Math.pow(t_1, 3.0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	tmp = 0
	if math.hypot(1.0, x) <= 1.002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((-0.056243896484375 * math.pow(x, 8.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0))))
	else:
		tmp = ((2.0 - math.pow(t_1, 4.5)) + -1.0) / ((math.sqrt(t_1) + (t_0 + 1.5)) * (1.0 + (math.pow(t_1, 1.5) + math.pow(t_1, 3.0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(-0.056243896484375 * (x ^ 8.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0)))));
	else
		tmp = Float64(Float64(Float64(2.0 - (t_1 ^ 4.5)) + -1.0) / Float64(Float64(sqrt(t_1) + Float64(t_0 + 1.5)) * Float64(1.0 + Float64((t_1 ^ 1.5) + (t_1 ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = 0.5 + t_0;
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((-0.056243896484375 * (x ^ 8.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0))));
	else
		tmp = ((2.0 - (t_1 ^ 4.5)) + -1.0) / ((sqrt(t_1) + (t_0 + 1.5)) * (1.0 + ((t_1 ^ 1.5) + (t_1 ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.056243896484375 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 - N[Power[t$95$1, 4.5], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(N[Sqrt[t$95$1], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[t$95$1, 1.5], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.002

   1. Initial program 57.3%

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

      1. distribute-lft-in 57.3%

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

      2. metadata-eval 57.3%

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

      3. associate-*r/ 57.3%

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

      4. metadata-eval 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)} \]

   if 1.002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip3-- 97.5%

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

      2. div-inv 97.5%

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

      3. metadata-eval 97.5%

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

      4. sqrt-pow2 98.4%

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

      5. metadata-eval 98.4%

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

      6. metadata-eval 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. *-un-lft-identity 98.4%

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

      9. associate-+r+ 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

   8. Simplified 99.9%

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

      1. sub-neg 99.9%

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

      2. flip3-+ 98.4%

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

      3. metadata-eval 98.4%

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

      4. metadata-eval 98.4%

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

      5. *-un-lft-identity 98.4%

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

   10. Applied egg-rr 98.4%

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

       1. cube-neg 98.4%

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

       2. unsub-neg 98.4%

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

       3. associate-+r- 99.9%

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

       4. sub-neg 99.9%

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

       5. remove-double-neg 99.9%

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

       6. associate-+r+ 98.4%

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

       7. +-commutative 98.4%

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

   12. Simplified 99.9%

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

       1. expm1-log1p-u 98.4%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}\right)}^{3}\right)\right)}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       2. pow-pow 98.4%

          \[\leadsto \frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \color{blue}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\left(1.5 \cdot 3\right)}}\right)\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       3. metadata-eval 98.4%

          \[\leadsto \frac{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{\color{blue}{4.5}}\right)\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

   14. Applied egg-rr 98.4%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)\right)}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]
   15. Step-by-step derivation

       1. expm1-undefine 98.4%

          \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)} - 1}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       2. sub-neg 98.4%

          \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)} + \left(-1\right)}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       3. log1p-undefine 99.9%

          \[\leadsto \frac{\frac{e^{\color{blue}{\log \left(1 + \left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)\right)}} + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       4. rem-exp-log 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\left(1 + \left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)\right)} + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       5. associate-+r- 99.9%

          \[\leadsto \frac{\frac{\color{blue}{\left(\left(1 + 1\right) - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)} + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       6. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\left(\color{blue}{2} - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + \left(-1\right)}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + \color{blue}{-1}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]

   16. Simplified 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)} \]
   17. Step-by-step derivation

       1. *-un-lft-identity 99.9%

          \[\leadsto \color{blue}{1 \cdot \frac{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}} \]

       2. associate-/l/ 99.9%

          \[\leadsto 1 \cdot \color{blue}{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}} \]

   18. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{1 \cdot \frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}} \]
   19. Step-by-step derivation

       1. *-lft-identity 99.9%

          \[\leadsto \color{blue}{\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}} \]

       2. +-commutative 99.9%

          \[\leadsto \frac{\color{blue}{-1 + \left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)}}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)} \]

       3. *-commutative 99.9%

          \[\leadsto \frac{-1 + \left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)}{\color{blue}{\left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right) \cdot \left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right)}} \]

       4. +-commutative 99.9%

          \[\leadsto \frac{-1 + \left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)}{\left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right) \cdot \left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \color{blue}{\left(1.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)} \]

   20. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{-1 + \left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right)}{\left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right) \cdot \left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(1.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4.5}\right) + -1}{\left(\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)\right) \cdot \left(1 + \left({\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5} + {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_1}^{1.5}}{\sqrt{t\_1} + \left(t\_0 + 1.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+
       (* -0.056243896484375 (pow x 8.0))
       (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0)))))
     (/ (- 1.0 (pow t_1 1.5)) (+ (sqrt t_1) (+ t_0 1.5))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((-0.056243896484375 * pow(x, 8.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0))));
	} else {
		tmp = (1.0 - pow(t_1, 1.5)) / (sqrt(t_1) + (t_0 + 1.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double t_1 = 0.5 + t_0;
	double tmp;
	if (Math.hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((-0.056243896484375 * Math.pow(x, 8.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0))));
	} else {
		tmp = (1.0 - Math.pow(t_1, 1.5)) / (Math.sqrt(t_1) + (t_0 + 1.5));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	t_1 = 0.5 + t_0
	tmp = 0
	if math.hypot(1.0, x) <= 1.002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((-0.056243896484375 * math.pow(x, 8.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0))))
	else:
		tmp = (1.0 - math.pow(t_1, 1.5)) / (math.sqrt(t_1) + (t_0 + 1.5))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 + t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(-0.056243896484375 * (x ^ 8.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0)))));
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 1.5)) / Float64(sqrt(t_1) + Float64(t_0 + 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	t_1 = 0.5 + t_0;
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((-0.056243896484375 * (x ^ 8.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0))));
	else
		tmp = (1.0 - (t_1 ^ 1.5)) / (sqrt(t_1) + (t_0 + 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.056243896484375 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.002

   1. Initial program 57.3%

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

      1. distribute-lft-in 57.3%

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

      2. metadata-eval 57.3%

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

      3. associate-*r/ 57.3%

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

      4. metadata-eval 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)} \]

   if 1.002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip3-- 97.5%

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

      2. div-inv 97.5%

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

      3. metadata-eval 97.5%

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

      4. sqrt-pow2 98.4%

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

      5. metadata-eval 98.4%

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

      6. metadata-eval 98.4%

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

      7. add-sqr-sqrt 98.4%

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

      8. *-un-lft-identity 98.4%

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

      9. associate-+r+ 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. metadata-eval 99.9%

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

      6. +-commutative 99.9%

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

   8. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{1.5}}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+
       (* -0.056243896484375 (pow x 8.0))
       (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0)))))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((-0.056243896484375 * pow(x, 8.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0))));
	} 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) <= 1.002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((-0.056243896484375 * Math.pow(x, 8.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0))));
	} 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) <= 1.002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((-0.056243896484375 * math.pow(x, 8.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0))))
	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) <= 1.002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(-0.056243896484375 * (x ^ 8.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0)))));
	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) <= 1.002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((-0.056243896484375 * (x ^ 8.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0))));
	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], 1.002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.056243896484375 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.002

   1. Initial program 57.3%

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

      1. distribute-lft-in 57.3%

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

      2. metadata-eval 57.3%

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

      3. associate-*r/ 57.3%

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

      4. metadata-eval 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)} \]

   if 1.002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(-0.056243896484375 \cdot {x}^{8} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.002)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} 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) <= 1.002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} 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) <= 1.002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	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) <= 1.002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	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) <= 1.002)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	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], 1.002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.002

   1. Initial program 57.3%

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

      1. distribute-lft-in 57.3%

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

      2. metadata-eval 57.3%

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

      3. associate-*r/ 57.3%

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

      4. metadata-eval 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)} \]

   if 1.002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.5)
   (+
    (* -0.0859375 (pow x 4.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
   (/ (+ 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.5) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.5) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = (0.5 + (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.5:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = (0.5 + (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.5)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.5)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.5

   1. Initial program 57.6%

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

      1. distribute-lft-in 57.6%

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

      2. metadata-eval 57.6%

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

      3. associate-*r/ 57.6%

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

      4. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)} \]

   if 1.5 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 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%

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

      1. flip-- 96.8%

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

      2. div-inv 96.8%

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

      3. metadata-eval 96.8%

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

      4. add-sqr-sqrt 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. associate--r+ 98.3%

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

       2. metadata-eval 98.3%

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

       3. associate-*r/ 98.3%

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

       4. *-rgt-identity 98.3%

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

       5. sub-neg 98.3%

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

       6. distribute-neg-frac 98.3%

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

       7. metadata-eval 98.3%

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

   11. Simplified 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.5)
   (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
   (/ (+ 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.5) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.5) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * Math.pow(x, 2.0));
	} else {
		tmp = (0.5 + (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.5:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * math.pow(x, 2.0))
	else:
		tmp = (0.5 + (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.5)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.5)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.5

   1. Initial program 57.6%

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

      1. distribute-lft-in 57.6%

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

      2. metadata-eval 57.6%

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

      3. associate-*r/ 57.6%

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

      4. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]

   if 1.5 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 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%

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

      1. flip-- 96.8%

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

      2. div-inv 96.8%

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

      3. metadata-eval 96.8%

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

      4. add-sqr-sqrt 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. associate--r+ 98.3%

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

       2. metadata-eval 98.3%

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

       3. associate-*r/ 98.3%

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

       4. *-rgt-identity 98.3%

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

       5. sub-neg 98.3%

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

       6. distribute-neg-frac 98.3%

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

       7. metadata-eval 98.3%

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

   11. Simplified 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.5

   1. Initial program 57.6%

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

      1. distribute-lft-in 57.6%

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

      2. metadata-eval 57.6%

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

      3. associate-*r/ 57.6%

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

      4. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 1.5 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 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%

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

      1. flip-- 96.8%

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

      2. div-inv 96.8%

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

      3. metadata-eval 96.8%

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

      4. add-sqr-sqrt 98.2%

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

   9. Applied egg-rr 98.2%

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

       1. associate--r+ 98.3%

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

       2. metadata-eval 98.3%

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

       3. associate-*r/ 98.3%

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

       4. *-rgt-identity 98.3%

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

       5. sub-neg 98.3%

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

       6. distribute-neg-frac 98.3%

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

       7. metadata-eval 98.3%

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

   11. Simplified 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.5) {
		tmp = 0.125 * pow(x, 2.0);
	} 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) <= 1.5) {
		tmp = 0.125 * Math.pow(x, 2.0);
	} 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) <= 1.5:
		tmp = 0.125 * math.pow(x, 2.0)
	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) <= 1.5)
		tmp = Float64(0.125 * (x ^ 2.0));
	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) <= 1.5)
		tmp = 0.125 * (x ^ 2.0);
	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], 1.5], N[(0.125 * N[Power[x, 2.0], $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 1 x) < 1.5

   1. Initial program 57.6%

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

      1. distribute-lft-in 57.6%

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

      2. metadata-eval 57.6%

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

      3. associate-*r/ 57.6%

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

      4. metadata-eval 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 1.5 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 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 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (* 0.125 (pow x 2.0)) (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * pow(x, 2.0);
	} 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 ** 2.0d0)
    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 * Math.pow(x, 2.0);
	} 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 * math.pow(x, 2.0)
	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 * (x ^ 2.0));
	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 ^ 2.0);
	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[Power[x, 2.0], $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 73.1%

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

      1. distribute-lft-in 73.1%

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

      2. metadata-eval 73.1%

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

      3. associate-*r/ 73.1%

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

      4. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around 0 63.1%

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

   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 96.9%

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

      1. flip-- 96.9%

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

      2. metadata-eval 96.9%

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

      3. pow1/2 96.9%

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

      4. pow1/2 96.9%

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

      5. pow-prod-up 98.3%

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

      6. metadata-eval 98.3%

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

      7. metadata-eval 98.3%

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

      8. metadata-eval 98.3%

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

   7. Applied egg-rr 98.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (* 0.125 (pow x 2.0)) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = 0.125d0 * (x ** 2.0d0)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * Math.pow(x, 2.0);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = 0.125 * math.pow(x, 2.0)
	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 * (x ^ 2.0));
	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 ^ 2.0);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(0.125 * N[Power[x, 2.0], $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 73.1%

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

      1. distribute-lft-in 73.1%

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

      2. metadata-eval 73.1%

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

      3. associate-*r/ 73.1%

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

      4. metadata-eval 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in x around 0 63.1%

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

   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 96.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.2e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   1. Initial program 77.1%

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

      1. distribute-lft-in 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-*r/ 77.1%

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

      4. metadata-eval 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in x around 0 38.1%

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

   if 2.20000000000000007e-77 < x

   1. Initial program 83.5%

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

      1. distribute-lft-in 83.5%

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

      2. metadata-eval 83.5%

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

      3. associate-*r/ 83.5%

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

      4. metadata-eval 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x around inf 81.5%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 32.0% accurate, 34.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3e-77) 0.0 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 3e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3e-77:
		tmp = 0.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e-77)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3e-77)
		tmp = 0.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3e-77], 0.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.00000000000000016e-77

   1. Initial program 77.1%

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

      1. distribute-lft-in 77.1%

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

      2. metadata-eval 77.1%

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

      3. associate-*r/ 77.1%

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

      4. metadata-eval 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in x around 0 38.1%

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

   if 3.00000000000000016e-77 < x

   1. Initial program 83.5%

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

      1. distribute-lft-in 83.5%

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

      2. metadata-eval 83.5%

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

      3. associate-*r/ 83.5%

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

      4. metadata-eval 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x around inf 81.9%

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

      1. associate-*r/ 81.9%

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

      2. metadata-eval 81.9%

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

   7. Simplified 81.9%

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

      1. expm1-log1p-u 81.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)\right)} \]

   9. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)\right)} \]
   10. Step-by-step derivation

       1. expm1-undefine 80.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)} - 1} \]

       2. sub-neg 80.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)} + \left(-1\right)} \]

       3. log1p-undefine 81.6%

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

       4. rem-exp-log 81.9%

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

       5. associate-+r- 81.9%

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

       6. metadata-eval 81.9%

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

       7. metadata-eval 81.9%

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

   11. Simplified 81.9%

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

   12. Taylor expanded in x around 0 15.7%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 10.8% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 78.8%

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

   1. distribute-lft-in 78.8%

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

   2. metadata-eval 78.8%

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

   3. associate-*r/ 78.8%

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

   4. metadata-eval 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in x around inf 50.9%

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

   1. associate-*r/ 50.9%

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

   2. metadata-eval 50.9%

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

7. Simplified 50.9%

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

   1. expm1-log1p-u 50.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)\right)} \]

9. Applied egg-rr 50.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)\right)} \]
10. Step-by-step derivation

    1. expm1-undefine 49.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)} - 1} \]

    2. sub-neg 49.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - \sqrt{0.5 + \frac{0.5}{x}}\right)} + \left(-1\right)} \]

    3. log1p-undefine 50.3%

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

    4. rem-exp-log 50.9%

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

    5. associate-+r- 50.9%

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

    6. metadata-eval 50.9%

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

    7. metadata-eval 50.9%

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

11. Simplified 50.9%

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

12. Taylor expanded in x around 0 11.0%

    \[\leadsto \color{blue}{2} + -1 \]

13. Final simplification 11.0%

    \[\leadsto 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024043 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))