Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.8%

Time: 25.3s

Alternatives: 15

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 1 + \sqrt{0.5 + t_0}\\ t_2 := \frac{0.5}{t_1}\\ t_3 := \frac{t_0}{t_1}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2 \cdot t_2 - t_3 \cdot t_3}{t_2 + t_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x)))
        (t_1 (+ 1.0 (sqrt (+ 0.5 t_0))))
        (t_2 (/ 0.5 t_1))
        (t_3 (/ t_0 t_1)))
   (if (<= (hypot 1.0 x) 1.02)
     (fma x (* x 0.125) (* (pow x 4.0) -0.0859375))
     (/ (- (* t_2 t_2) (* t_3 t_3)) (+ t_2 t_3)))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 1.0 + sqrt((0.5 + t_0));
	double t_2 = 0.5 / t_1;
	double t_3 = t_0 / t_1;
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = fma(x, (x * 0.125), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = ((t_2 * t_2) - (t_3 * t_3)) / (t_2 + t_3);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(1.0 + sqrt(Float64(0.5 + t_0)))
	t_2 = Float64(0.5 / t_1)
	t_3 = Float64(t_0 / t_1)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = fma(x, Float64(x * 0.125), Float64((x ^ 4.0) * -0.0859375));
	else
		tmp = Float64(Float64(Float64(t_2 * t_2) - Float64(t_3 * t_3)) / Float64(t_2 + t_3));
	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[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * 0.125), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. pow3 49.2%

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

      4. pow-pow 49.1%

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

      5. metadata-eval 49.1%

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

   5. Applied egg-rr 49.1%

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

      1. pow1/3 49.2%

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

      2. pow-pow 49.2%

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

      3. metadata-eval 49.2%

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

      4. pow1/2 49.2%

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

      5. flip-- 49.2%

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

      6. metadata-eval 49.2%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate--r+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. div-sub 49.2%

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

      11. flip-- 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, -0.0859375 \cdot {x}^{4}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.125, \color{blue}{{x}^{4} \cdot -0.0859375}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)} \]

   if 1.02 < (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. pow1/2 98.4%

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

      2. add-cube-cbrt 98.3%

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

      3. pow3 98.3%

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

      4. pow-pow 98.3%

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

      5. metadata-eval 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. pow1/3 98.4%

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

      2. pow-pow 98.4%

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

      3. metadata-eval 98.4%

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

      4. pow1/2 98.4%

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

      5. flip-- 98.4%

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

      6. metadata-eval 98.4%

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

      7. add-sqr-sqrt 99.9%

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

      8. associate--r+ 99.9%

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

      9. metadata-eval 99.9%

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

      10. div-sub 100.0%

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

      11. flip-- 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} - \frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{\frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} + \frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 1 + \sqrt{0.5 + t_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{{t_1}^{2}} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right) \cdot t_1}\right)}^{2}}{\frac{0.5}{t_1} + \frac{t_0}{t_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))) (t_1 (+ 1.0 (sqrt (+ 0.5 t_0)))))
   (if (<= (hypot 1.0 x) 1.02)
     (fma x (* x 0.125) (* (pow x 4.0) -0.0859375))
     (/
      (- (/ 0.25 (pow t_1 2.0)) (pow (/ 0.5 (* (hypot 1.0 x) t_1)) 2.0))
      (+ (/ 0.5 t_1) (/ t_0 t_1))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double t_1 = 1.0 + sqrt((0.5 + t_0));
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = fma(x, (x * 0.125), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = ((0.25 / pow(t_1, 2.0)) - pow((0.5 / (hypot(1.0, x) * t_1)), 2.0)) / ((0.5 / t_1) + (t_0 / t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(1.0 + sqrt(Float64(0.5 + t_0)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = fma(x, Float64(x * 0.125), Float64((x ^ 4.0) * -0.0859375));
	else
		tmp = Float64(Float64(Float64(0.25 / (t_1 ^ 2.0)) - (Float64(0.5 / Float64(hypot(1.0, x) * t_1)) ^ 2.0)) / Float64(Float64(0.5 / t_1) + Float64(t_0 / t_1)));
	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[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * 0.125), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[N[(0.5 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 / t$95$1), $MachinePrecision] + N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. pow3 49.2%

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

      4. pow-pow 49.1%

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

      5. metadata-eval 49.1%

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

   5. Applied egg-rr 49.1%

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

      1. pow1/3 49.2%

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

      2. pow-pow 49.2%

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

      3. metadata-eval 49.2%

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

      4. pow1/2 49.2%

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

      5. flip-- 49.2%

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

      6. metadata-eval 49.2%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate--r+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. div-sub 49.2%

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

      11. flip-- 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, -0.0859375 \cdot {x}^{4}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.125, \color{blue}{{x}^{4} \cdot -0.0859375}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)} \]

   if 1.02 < (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. pow1/2 98.4%

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

      2. add-cube-cbrt 98.3%

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

      3. pow3 98.3%

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

      4. pow-pow 98.3%

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

      5. metadata-eval 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. pow1/3 98.4%

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

      2. pow-pow 98.4%

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

      3. metadata-eval 98.4%

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

      4. pow1/2 98.4%

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

      5. flip-- 98.4%

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

      6. metadata-eval 98.4%

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

      7. add-sqr-sqrt 99.9%

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

      8. associate--r+ 99.9%

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

      9. metadata-eval 99.9%

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

      10. div-sub 100.0%

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

      11. flip-- 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. sub-neg 100.0%

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

      2. frac-times 99.9%

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

      3. metadata-eval 99.9%

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

      4. pow2 99.9%

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

      5. pow2 99.9%

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

      6. associate-/l/ 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. unsub-neg 99.9%

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

       2. *-commutative 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2}} - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right) \cdot \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\right)}^{2}}{\frac{0.5}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} + \frac{\frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{t_0}}{1 + \sqrt{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.02)
     (fma x (* x 0.125) (* (pow x 4.0) -0.0859375))
     (/ (/ (- 0.25 (/ 0.25 (+ 1.0 (* x x)))) t_0) (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = fma(x, (x * 0.125), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = ((0.25 - (0.25 / (1.0 + (x * x)))) / t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = fma(x, Float64(x * 0.125), Float64((x ^ 4.0) * -0.0859375));
	else
		tmp = Float64(Float64(Float64(0.25 - Float64(0.25 / Float64(1.0 + Float64(x * x)))) / t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * 0.125), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 - N[(0.25 / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. pow3 49.2%

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

      4. pow-pow 49.1%

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

      5. metadata-eval 49.1%

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

   5. Applied egg-rr 49.1%

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

      1. pow1/3 49.2%

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

      2. pow-pow 49.2%

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

      3. metadata-eval 49.2%

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

      4. pow1/2 49.2%

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

      5. flip-- 49.2%

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

      6. metadata-eval 49.2%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate--r+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. div-sub 49.2%

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

      11. flip-- 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, -0.0859375 \cdot {x}^{4}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.125, \color{blue}{{x}^{4} \cdot -0.0859375}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)} \]

   if 1.02 < (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-- 98.4%

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

      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%

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

   5. Applied egg-rr 99.9%

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

      1. flip-- 22.8%

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

      2. div-inv 22.8%

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

      3. metadata-eval 22.8%

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

      4. frac-times 22.8%

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

      5. metadata-eval 22.8%

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

      6. hypot-udef 22.8%

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

      7. hypot-udef 22.8%

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

      8. add-sqr-sqrt 22.8%

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

      9. metadata-eval 22.8%

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r/ 22.8%

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

      2. *-rgt-identity 22.8%

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

      3. unpow2 22.8%

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

      4. +-commutative 22.8%

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

      5. unpow2 22.8%

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

   9. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t_0}{1 + \sqrt{0.5 + t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.02)
     (fma x (* x 0.125) (* (pow x 4.0) -0.0859375))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = fma(x, (x * 0.125), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = (0.5 - t_0) / (1.0 + 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) <= 1.02)
		tmp = fma(x, Float64(x * 0.125), Float64((x ^ 4.0) * -0.0859375));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(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]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * 0.125), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $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) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. pow3 49.2%

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

      4. pow-pow 49.1%

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

      5. metadata-eval 49.1%

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

   5. Applied egg-rr 49.1%

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

      1. pow1/3 49.2%

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

      2. pow-pow 49.2%

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

      3. metadata-eval 49.2%

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

      4. pow1/2 49.2%

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

      5. flip-- 49.2%

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

      6. metadata-eval 49.2%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate--r+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. div-sub 49.2%

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

      11. flip-- 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, -0.0859375 \cdot {x}^{4}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.125, \color{blue}{{x}^{4} \cdot -0.0859375}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)} \]

   if 1.02 < (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-- 98.4%

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

      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%

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

   5. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 5: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.02)
   (* x (* x 0.125))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.02) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.02:
		tmp = x * (x * 0.125)
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.02)
		tmp = x * (x * 0.125);
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. pow3 49.2%

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

      4. pow-pow 49.1%

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

      5. metadata-eval 49.1%

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

   5. Applied egg-rr 49.1%

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

      1. pow1/3 49.2%

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

      2. pow-pow 49.2%

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

      3. metadata-eval 49.2%

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

      4. pow1/2 49.2%

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

      5. flip-- 49.2%

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

      6. metadata-eval 49.2%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate--r+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. div-sub 49.2%

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

      11. flip-- 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*l* 99.8%

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

   10. Simplified 99.8%

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

   if 1.02 < (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%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 6: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.02)
   (fma 0.125 (* x x) (* (pow x 4.0) -0.0859375))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = fma(0.125, (x * x), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = fma(0.125, Float64(x * x), Float64((x ^ 4.0) * -0.0859375));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(0.125 * N[(x * x), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   5. Step-by-step derivation

      1. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, -0.0859375 \cdot {x}^{4}\right)} \]

      2. unpow2 100.0%

         \[\leadsto \mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, -0.0859375 \cdot {x}^{4}\right) \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x \cdot x, -0.0859375 \cdot {x}^{4}\right)} \]

   if 1.02 < (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%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(0.125, x \cdot x, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 7: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.02)
   (fma x (* x 0.125) (* (pow x 4.0) -0.0859375))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = fma(x, (x * 0.125), (pow(x, 4.0) * -0.0859375));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = fma(x, Float64(x * 0.125), Float64((x ^ 4.0) * -0.0859375));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(x * N[(x * 0.125), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * -0.0859375), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.02

   1. Initial program 49.2%

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

      1. distribute-lft-in 49.2%

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

      2. metadata-eval 49.2%

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

      3. associate-*r/ 49.2%

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

      4. metadata-eval 49.2%

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

   3. Simplified 49.2%

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

      1. pow1/2 49.2%

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

      2. add-cube-cbrt 49.2%

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

      3. pow3 49.2%

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

      4. pow-pow 49.1%

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

      5. metadata-eval 49.1%

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

   5. Applied egg-rr 49.1%

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

      1. pow1/3 49.2%

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

      2. pow-pow 49.2%

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

      3. metadata-eval 49.2%

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

      4. pow1/2 49.2%

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

      5. flip-- 49.2%

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

      6. metadata-eval 49.2%

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

      7. add-sqr-sqrt 49.2%

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

      8. associate--r+ 49.2%

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

      9. metadata-eval 49.2%

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

      10. div-sub 49.2%

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

      11. flip-- 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + -0.0859375 \cdot {x}^{4}} \]
   9. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, -0.0859375 \cdot {x}^{4}\right)} \]

      5. *-commutative 100.0%

         \[\leadsto \mathsf{fma}\left(x, x \cdot 0.125, \color{blue}{{x}^{4} \cdot -0.0859375}\right) \]

   10. Simplified 100.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)} \]

   if 1.02 < (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%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, {x}^{4} \cdot -0.0859375\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 8: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. add-cube-cbrt 49.5%

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

      3. pow3 49.5%

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

      4. pow-pow 49.5%

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

      5. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. pow1/3 49.6%

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

      2. pow-pow 49.6%

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

      3. metadata-eval 49.6%

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

      4. pow1/2 49.6%

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

      5. flip-- 49.6%

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

      6. metadata-eval 49.6%

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

      7. add-sqr-sqrt 49.6%

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

      8. associate--r+ 49.6%

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

      9. metadata-eval 49.6%

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

      10. div-sub 49.6%

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

      11. flip-- 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

   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. Taylor expanded in x around inf 96.0%

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

      1. flip-- 96.0%

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

      2. metadata-eval 96.0%

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

      3. add-sqr-sqrt 97.5%

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

      4. associate--r+ 97.5%

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

      5. metadata-eval 97.5%

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

   6. Applied egg-rr 97.5%

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

4. Final simplification 98.3%

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

Alternative 9: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. add-cube-cbrt 49.5%

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

      3. pow3 49.5%

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

      4. pow-pow 49.5%

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

      5. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. pow1/3 49.6%

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

      2. pow-pow 49.6%

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

      3. metadata-eval 49.6%

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

      4. pow1/2 49.6%

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

      5. flip-- 49.6%

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

      6. metadata-eval 49.6%

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

      7. add-sqr-sqrt 49.6%

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

      8. associate--r+ 49.6%

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

      9. metadata-eval 49.6%

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

      10. div-sub 49.6%

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

      11. flip-- 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

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

   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. 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 97.0%

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

4. Final simplification 98.1%

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

Alternative 10: 97.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004 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 95.6%

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

   if -1.55000000000000004 < x < 1.55000000000000004

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. add-cube-cbrt 49.5%

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

      3. pow3 49.5%

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

      4. pow-pow 49.5%

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

      5. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. pow1/3 49.6%

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

      2. pow-pow 49.6%

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

      3. metadata-eval 49.6%

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

      4. pow1/2 49.6%

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

      5. flip-- 49.6%

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

      6. metadata-eval 49.6%

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

      7. add-sqr-sqrt 49.6%

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

      8. associate--r+ 49.6%

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

      9. metadata-eval 49.6%

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

      10. div-sub 49.6%

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

      11. flip-- 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

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

Alternative 11: 61.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{0.5 + \frac{0.5}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.85) (not (<= x 0.65)))
   (/ (/ (- 0.25 (/ 0.25 (+ 1.0 (* x x)))) (+ 0.5 (/ 0.5 x))) 2.0)
   (* x (* x 0.125))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.85) || !(x <= 0.65)) {
		tmp = ((0.25 - (0.25 / (1.0 + (x * x)))) / (0.5 + (0.5 / x))) / 2.0;
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.85d0)) .or. (.not. (x <= 0.65d0))) then
        tmp = ((0.25d0 - (0.25d0 / (1.0d0 + (x * x)))) / (0.5d0 + (0.5d0 / x))) / 2.0d0
    else
        tmp = x * (x * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.85) || !(x <= 0.65)) {
		tmp = ((0.25 - (0.25 / (1.0 + (x * x)))) / (0.5 + (0.5 / x))) / 2.0;
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.85) or not (x <= 0.65):
		tmp = ((0.25 - (0.25 / (1.0 + (x * x)))) / (0.5 + (0.5 / x))) / 2.0
	else:
		tmp = x * (x * 0.125)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.85) || !(x <= 0.65))
		tmp = Float64(Float64(Float64(0.25 - Float64(0.25 / Float64(1.0 + Float64(x * x)))) / Float64(0.5 + Float64(0.5 / x))) / 2.0);
	else
		tmp = Float64(x * Float64(x * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.85) || ~((x <= 0.65)))
		tmp = ((0.25 - (0.25 / (1.0 + (x * x)))) / (0.5 + (0.5 / x))) / 2.0;
	else
		tmp = x * (x * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.85], N[Not[LessEqual[x, 0.65]], $MachinePrecision]], N[(N[(N[(0.25 - N[(0.25 / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8500000000000001 or 0.650000000000000022 < 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 0 22.7%

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

      1. flip-- 22.7%

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

      2. div-inv 22.7%

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

      3. metadata-eval 22.7%

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

      4. frac-times 22.7%

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

      5. metadata-eval 22.7%

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

      6. hypot-udef 22.7%

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

      7. hypot-udef 22.7%

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

      8. add-sqr-sqrt 22.7%

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

      9. metadata-eval 22.7%

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

   8. Applied egg-rr 22.7%

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

      1. associate-*r/ 22.7%

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

      2. *-rgt-identity 22.7%

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

      3. unpow2 22.7%

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

      4. +-commutative 22.7%

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

      5. unpow2 22.7%

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

   10. Simplified 22.7%

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

   11. Taylor expanded in x around inf 22.7%

       \[\leadsto \frac{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{0.5 + \color{blue}{\frac{0.5}{x}}}}{2} \]

   if -1.8500000000000001 < x < 0.650000000000000022

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. add-cube-cbrt 49.5%

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

      3. pow3 49.5%

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

      4. pow-pow 49.5%

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

      5. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. pow1/3 49.6%

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

      2. pow-pow 49.6%

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

      3. metadata-eval 49.6%

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

      4. pow1/2 49.6%

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

      5. flip-- 49.6%

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

      6. metadata-eval 49.6%

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

      7. add-sqr-sqrt 49.6%

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

      8. associate--r+ 49.6%

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

      9. metadata-eval 49.6%

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

      10. div-sub 49.6%

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

      11. flip-- 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{0.5 + \frac{0.5}{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 12: 61.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.8) (not (<= x 1.25))) (- 0.25 (/ 0.25 x)) (* x (* x 0.125))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.8) || !(x <= 1.25)) {
		tmp = 0.25 - (0.25 / x);
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.8d0)) .or. (.not. (x <= 1.25d0))) then
        tmp = 0.25d0 - (0.25d0 / x)
    else
        tmp = x * (x * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.8) || !(x <= 1.25)) {
		tmp = 0.25 - (0.25 / x);
	} else {
		tmp = x * (x * 0.125);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.8) or not (x <= 1.25):
		tmp = 0.25 - (0.25 / x)
	else:
		tmp = x * (x * 0.125)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.8) || !(x <= 1.25))
		tmp = Float64(0.25 - Float64(0.25 / x));
	else
		tmp = Float64(x * Float64(x * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.8) || ~((x <= 1.25)))
		tmp = 0.25 - (0.25 / x);
	else
		tmp = x * (x * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.8], N[Not[LessEqual[x, 1.25]], $MachinePrecision]], N[(0.25 - N[(0.25 / x), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004 or 1.25 < 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 0 22.7%

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

   7. Taylor expanded in x around inf 22.7%

      \[\leadsto \color{blue}{0.25 - 0.25 \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. associate-*r/ 22.7%

         \[\leadsto 0.25 - \color{blue}{\frac{0.25 \cdot 1}{x}} \]

      2. metadata-eval 22.7%

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

   9. Simplified 22.7%

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

   if -1.80000000000000004 < x < 1.25

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. add-cube-cbrt 49.5%

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

      3. pow3 49.5%

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

      4. pow-pow 49.5%

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

      5. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. pow1/3 49.6%

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

      2. pow-pow 49.6%

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

      3. metadata-eval 49.6%

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

      4. pow1/2 49.6%

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

      5. flip-- 49.6%

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

      6. metadata-eval 49.6%

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

      7. add-sqr-sqrt 49.6%

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

      8. associate--r+ 49.6%

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

      9. metadata-eval 49.6%

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

      10. div-sub 49.6%

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

      11. flip-- 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;0.25 - \frac{0.25}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 13: 61.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.45) 0.25 (if (<= x 1.42) (* 0.125 (* x x)) 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.42) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.45d0)) then
        tmp = 0.25d0
    else if (x <= 1.42d0) then
        tmp = 0.125d0 * (x * x)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.42) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.45:
		tmp = 0.25
	elif x <= 1.42:
		tmp = 0.125 * (x * x)
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.42)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.42)
		tmp = 0.125 * (x * x);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.45], 0.25, If[LessEqual[x, 1.42], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.4199999999999999 < 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 0 22.7%

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

   7. Taylor expanded in x around inf 22.7%

      \[\leadsto \frac{\color{blue}{0.5}}{2} \]

   if -1.44999999999999996 < x < 1.4199999999999999

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

   4. Taylor expanded in x around 0 99.2%

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

      1. unpow2 99.2%

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

   6. Simplified 99.2%

      \[\leadsto \color{blue}{0.125 \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 14: 61.1% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.45) 0.25 (if (<= x 1.42) (* x (* x 0.125)) 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.42) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.45d0)) then
        tmp = 0.25d0
    else if (x <= 1.42d0) then
        tmp = x * (x * 0.125d0)
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.25;
	} else if (x <= 1.42) {
		tmp = x * (x * 0.125);
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.45:
		tmp = 0.25
	elif x <= 1.42:
		tmp = x * (x * 0.125)
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.42)
		tmp = Float64(x * Float64(x * 0.125));
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = 0.25;
	elseif (x <= 1.42)
		tmp = x * (x * 0.125);
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.45], 0.25, If[LessEqual[x, 1.42], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision], 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.4199999999999999 < 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 0 22.7%

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

   7. Taylor expanded in x around inf 22.7%

      \[\leadsto \frac{\color{blue}{0.5}}{2} \]

   if -1.44999999999999996 < x < 1.4199999999999999

   1. Initial program 49.6%

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

      1. distribute-lft-in 49.6%

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

      2. metadata-eval 49.6%

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

      3. associate-*r/ 49.6%

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

      4. metadata-eval 49.6%

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

   3. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. add-cube-cbrt 49.5%

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

      3. pow3 49.5%

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

      4. pow-pow 49.5%

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

      5. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. pow1/3 49.6%

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

      2. pow-pow 49.6%

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

      3. metadata-eval 49.6%

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

      4. pow1/2 49.6%

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

      5. flip-- 49.6%

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

      6. metadata-eval 49.6%

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

      7. add-sqr-sqrt 49.6%

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

      8. associate--r+ 49.6%

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

      9. metadata-eval 49.6%

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

      10. div-sub 49.6%

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

      11. flip-- 49.6%

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

   7. Applied egg-rr 49.6%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*l* 99.2%

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

   10. Simplified 99.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 15: 13.4% accurate, 210.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

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

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 74.8%

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

   1. distribute-lft-in 74.8%

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

   2. metadata-eval 74.8%

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

   3. associate-*r/ 74.8%

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

   4. metadata-eval 74.8%

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

3. Simplified 74.8%

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

   1. flip-- 74.8%

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

   2. metadata-eval 74.8%

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

   3. add-sqr-sqrt 75.6%

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

   4. associate--r+ 75.6%

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

   5. metadata-eval 75.6%

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

5. Applied egg-rr 75.6%

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

6. Taylor expanded in x around 0 35.5%

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

7. Taylor expanded in x around inf 13.8%

   \[\leadsto \frac{\color{blue}{0.5}}{2} \]

8. Final simplification 13.8%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023274 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))