Given's Rotation SVD example, simplified

Percentage Accurate: 76.4% → 99.9%

Time: 10.1s

Alternatives: 10

Speedup: 1.9×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {t_1}^{3}\right) \cdot \frac{1}{{t_1}^{2} + \left(t_0 + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - t_0}}}\\ \end{array} \end{array} \]

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) 1.0002)
     (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
     (/
      (* (- 1.0 (pow t_1 3.0)) (/ 1.0 (+ (pow t_1 2.0) (+ t_0 1.5))))
      (+ 1.0 (/ (sqrt (- 0.25 (/ 0.25 (fma x x 1.0)))) (sqrt (- 0.5 t_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) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = ((1.0 - pow(t_1, 3.0)) * (1.0 / (pow(t_1, 2.0) + (t_0 + 1.5)))) / (1.0 + (sqrt((0.25 - (0.25 / fma(x, x, 1.0)))) / sqrt((0.5 - t_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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_1 ^ 3.0)) * Float64(1.0 / Float64((t_1 ^ 2.0) + Float64(t_0 + 1.5)))) / Float64(1.0 + Float64(sqrt(Float64(0.25 - Float64(0.25 / fma(x, x, 1.0)))) / sqrt(Float64(0.5 - t_0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 + t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Sqrt[N[(0.25 - N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

      3. associate-*r/ 51.3%

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

      4. metadata-eval 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. metadata-eval 99.8%

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

      2. associate--r+ 99.8%

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

      3. flip3-- 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. pow2 99.8%

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

      7. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

       1. div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

       1. flip-+ 99.9%

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

       2. sqrt-div 99.9%

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

       3. metadata-eval 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{\color{blue}{0.25} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       4. frac-times 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \color{blue}{\frac{0.5 \cdot 0.5}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       5. metadata-eval 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{\color{blue}{0.25}}{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       6. hypot-udef 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       7. hypot-udef 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       8. rem-square-sqrt 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\color{blue}{1 \cdot 1 + x \cdot x}}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       9. metadata-eval 99.9%

          \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\color{blue}{1} + x \cdot x}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       10. +-commutative 99.9%

           \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\color{blue}{x \cdot x + 1}}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

       11. fma-def 99.9%

           \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

   13. Applied egg-rr 99.9%

       \[\leadsto \frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \color{blue}{\frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \frac{\sqrt{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {t_1}^{3}\right) \cdot \frac{1}{{t_1}^{2} + \left(t_0 + 1.5\right)}}{1 + \sqrt{t_1}}\\ \end{array} \end{array} \]

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) 1.0002)
     (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
     (/
      (* (- 1.0 (pow t_1 3.0)) (/ 1.0 (+ (pow t_1 2.0) (+ t_0 1.5))))
      (+ 1.0 (sqrt t_1))))))

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) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = ((1.0 - pow(t_1, 3.0)) * (1.0 / (pow(t_1, 2.0) + (t_0 + 1.5)))) / (1.0 + sqrt(t_1));
	}
	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) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * Math.pow(x, 2.0));
	} else {
		tmp = ((1.0 - Math.pow(t_1, 3.0)) * (1.0 / (Math.pow(t_1, 2.0) + (t_0 + 1.5)))) / (1.0 + Math.sqrt(t_1));
	}
	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) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * math.pow(x, 2.0))
	else:
		tmp = ((1.0 - math.pow(t_1, 3.0)) * (1.0 / (math.pow(t_1, 2.0) + (t_0 + 1.5)))) / (1.0 + math.sqrt(t_1))
	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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_1 ^ 3.0)) * Float64(1.0 / Float64((t_1 ^ 2.0) + Float64(t_0 + 1.5)))) / Float64(1.0 + sqrt(t_1)));
	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) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = ((1.0 - (t_1 ^ 3.0)) * (1.0 / ((t_1 ^ 2.0) + (t_0 + 1.5)))) / (1.0 + sqrt(t_1));
	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], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

      3. associate-*r/ 51.3%

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

      4. metadata-eval 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. metadata-eval 99.8%

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

      2. associate--r+ 99.8%

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

      3. flip3-- 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. pow2 99.8%

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

      7. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

       1. div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}\right) \cdot \frac{1}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 + t_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {t_1}^{3}}{{t_1}^{2} + \left(t_0 + 1.5\right)}}{1 + \sqrt{t_1}}\\ \end{array} \end{array} \]

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) 1.0002)
     (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
     (/
      (/ (- 1.0 (pow t_1 3.0)) (+ (pow t_1 2.0) (+ t_0 1.5)))
      (+ 1.0 (sqrt t_1))))))

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) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = ((1.0 - pow(t_1, 3.0)) / (pow(t_1, 2.0) + (t_0 + 1.5))) / (1.0 + sqrt(t_1));
	}
	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) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * Math.pow(x, 2.0));
	} else {
		tmp = ((1.0 - Math.pow(t_1, 3.0)) / (Math.pow(t_1, 2.0) + (t_0 + 1.5))) / (1.0 + Math.sqrt(t_1));
	}
	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) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * math.pow(x, 2.0))
	else:
		tmp = ((1.0 - math.pow(t_1, 3.0)) / (math.pow(t_1, 2.0) + (t_0 + 1.5))) / (1.0 + math.sqrt(t_1))
	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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64((t_1 ^ 2.0) + Float64(t_0 + 1.5))) / Float64(1.0 + sqrt(t_1)));
	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) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = ((1.0 - (t_1 ^ 3.0)) / ((t_1 ^ 2.0) + (t_0 + 1.5))) / (1.0 + sqrt(t_1));
	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], 1.0002], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(t$95$0 + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

      3. associate-*r/ 51.3%

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

      4. metadata-eval 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. metadata-eval 99.8%

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

      2. associate--r+ 99.8%

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

      3. flip3-- 99.8%

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

      4. metadata-eval 99.8%

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

      5. metadata-eval 99.8%

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

      6. pow2 99.8%

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

      7. *-un-lft-identity 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+l+ 99.8%

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

      5. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + \left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 1.5\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0002)
     (+ (* -0.0859375 (pow x 4.0)) (* 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) <= 1.0002) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (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) <= 1.0002) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (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) <= 1.0002:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (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) <= 1.0002)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + 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) <= 1.0002)
		tmp = (-0.0859375 * (x ^ 4.0)) + (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], 1.0002], 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 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 51.3%

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

      1. distribute-lft-in 51.3%

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

      2. metadata-eval 51.3%

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

      3. associate-*r/ 51.3%

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

      4. metadata-eval 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.0002 < (hypot.f64 1 x)

   1. Initial program 98.4%

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

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

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

      1. flip-- 98.3%

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

      2. metadata-eval 98.3%

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

      3. add-sqr-sqrt 99.8%

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

      4. associate--r+ 99.8%

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

      5. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 5: 99.1% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+ (* -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) <= 2.0) {
		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) <= 2.0) {
		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) <= 2.0:
		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) <= 2.0)
		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) <= 2.0)
		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], 2.0], 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) < 2

   1. Initial program 51.5%

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

      1. distribute-lft-in 51.5%

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

      2. metadata-eval 51.5%

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

      3. associate-*r/ 51.5%

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

      4. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 99.7%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. flip-- 97.6%

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

      2. metadata-eval 97.6%

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

      3. add-sqr-sqrt 99.1%

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

      4. associate--r+ 99.1%

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

      5. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 99.4%

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

Alternative 6: 98.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* 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) <= 2.0) {
		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) <= 2.0) {
		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) <= 2.0:
		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) <= 2.0)
		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) <= 2.0)
		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], 2.0], 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) < 2

   1. Initial program 51.5%

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

      1. distribute-lft-in 51.5%

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

      2. metadata-eval 51.5%

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

      3. associate-*r/ 51.5%

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

      4. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 99.1%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. flip-- 97.6%

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

      2. metadata-eval 97.6%

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

      3. add-sqr-sqrt 99.1%

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

      4. associate--r+ 99.1%

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

      5. metadata-eval 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 99.1%

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

Alternative 7: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		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 (hypot(1.0, x) <= 2.0)
		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 (hypot(1.0, x) <= 2.0)
		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[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], 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 (hypot.f64 1 x) < 2

   1. Initial program 51.5%

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

      1. distribute-lft-in 51.5%

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

      2. metadata-eval 51.5%

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

      3. associate-*r/ 51.5%

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

      4. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 99.1%

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

   if 2 < (hypot.f64 1 x)

   1. Initial program 98.5%

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

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

      1. flip-- 98.5%

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

      2. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 97.7%

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

4. Final simplification 98.5%

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

Alternative 8: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.5) (not (<= x 1.55)))
   (- 1.0 (sqrt 0.5))
   (* 0.125 (pow x 2.0))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.55)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = 0.125 * pow(x, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.5d0)) .or. (.not. (x <= 1.55d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 0.125d0 * (x ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.5) || !(x <= 1.55)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = 0.125 * Math.pow(x, 2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.5) or not (x <= 1.55):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 0.125 * math.pow(x, 2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.5) || !(x <= 1.55))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(0.125 * (x ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.5) || ~((x <= 1.55)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = 0.125 * (x ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.5], N[Not[LessEqual[x, 1.55]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5 or 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. Taylor expanded in x around inf 96.3%

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

   if -1.5 < x < 1.55000000000000004

   1. Initial program 51.5%

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

      1. distribute-lft-in 51.5%

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

      2. metadata-eval 51.5%

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

      3. associate-*r/ 51.5%

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

      4. metadata-eval 51.5%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \end{array} \]

Alternative 9: 75.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.2e-77) (not (<= x 2.2e-77))) (- 1.0 (sqrt 0.5)) 0.0))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.2d-77)) .or. (.not. (x <= 2.2d-77))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -2.2e-77) or not (x <= 2.2e-77):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -2.2e-77], N[Not[LessEqual[x, 2.2e-77]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.20000000000000007e-77 or 2.20000000000000007e-77 < x

   1. Initial program 79.3%

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

      1. distribute-lft-in 79.3%

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

      2. metadata-eval 79.3%

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

      3. associate-*r/ 79.3%

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

      4. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in x around inf 77.0%

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

   if -2.20000000000000007e-77 < x < 2.20000000000000007e-77

   1. Initial program 62.2%

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

      1. distribute-lft-in 62.2%

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

      2. metadata-eval 62.2%

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

      3. associate-*r/ 62.2%

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

      4. metadata-eval 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in x around 0 62.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-77} \lor \neg \left(x \leq 2.2 \cdot 10^{-77}\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 27.8% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 71.5%

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

   1. distribute-lft-in 71.5%

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

   2. metadata-eval 71.5%

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

   3. associate-*r/ 71.5%

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

   4. metadata-eval 71.5%

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

3. Simplified 71.5%

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

4. Taylor expanded in x around 0 30.2%

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

5. Final simplification 30.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023315 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))