Given's Rotation SVD example, simplified

Percentage Accurate: 74.9% → 99.9%

Time: 5.2s

Alternatives: 17

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := t\_0 + 1\\ t_2 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_3 := \frac{1}{t\_2}\\ t_4 := \sqrt{2} \cdot {t\_2}^{2}\\ t_5 := \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{t\_4}, 0.1875 \cdot t\_3\right)\\ t_6 := 0.5 + t\_0 \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0085:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left({x\_m}^{2}, {x\_m}^{2} \cdot \left(0.15625 \cdot t\_3 - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{t\_4}, 0.25 \cdot \frac{\sqrt{0.5} \cdot t\_5}{\sqrt{2} \cdot t\_2}\right)\right) - t\_5, 0.25 \cdot t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {t\_6}^{4.5}}{1 + \mathsf{fma}\left({t\_1}^{3}, 0.125, {t\_6}^{1.5}\right)}}{1 + \mathsf{fma}\left(t\_1, 0.5, 1 \cdot \sqrt{t\_1 \cdot 0.5}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m)))
        (t_1 (+ t_0 1.0))
        (t_2 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_3 (/ 1.0 t_2))
        (t_4 (* (sqrt 2.0) (pow t_2 2.0)))
        (t_5 (fma -0.0625 (/ (sqrt 0.5) t_4) (* 0.1875 t_3)))
        (t_6 (+ 0.5 (* t_0 0.5))))
   (if (<= x_m 0.0085)
     (*
      (pow x_m 2.0)
      (fma
       (pow x_m 2.0)
       (-
        (*
         (pow x_m 2.0)
         (-
          (* 0.15625 t_3)
          (fma
           0.125
           (/
            (* (sqrt 0.5) (- 0.375 (* 0.0625 (/ 1.0 (pow (sqrt 2.0) 2.0)))))
            t_4)
           (* 0.25 (/ (* (sqrt 0.5) t_5) (* (sqrt 2.0) t_2))))))
        t_5)
       (* 0.25 t_3)))
     (/
      (/ (- 1.0 (pow t_6 4.5)) (+ 1.0 (fma (pow t_1 3.0) 0.125 (pow t_6 1.5))))
      (+ 1.0 (fma t_1 0.5 (* 1.0 (sqrt (* t_1 0.5)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = t_0 + 1.0;
	double t_2 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_3 = 1.0 / t_2;
	double t_4 = sqrt(2.0) * pow(t_2, 2.0);
	double t_5 = fma(-0.0625, (sqrt(0.5) / t_4), (0.1875 * t_3));
	double t_6 = 0.5 + (t_0 * 0.5);
	double tmp;
	if (x_m <= 0.0085) {
		tmp = pow(x_m, 2.0) * fma(pow(x_m, 2.0), ((pow(x_m, 2.0) * ((0.15625 * t_3) - fma(0.125, ((sqrt(0.5) * (0.375 - (0.0625 * (1.0 / pow(sqrt(2.0), 2.0))))) / t_4), (0.25 * ((sqrt(0.5) * t_5) / (sqrt(2.0) * t_2)))))) - t_5), (0.25 * t_3));
	} else {
		tmp = ((1.0 - pow(t_6, 4.5)) / (1.0 + fma(pow(t_1, 3.0), 0.125, pow(t_6, 1.5)))) / (1.0 + fma(t_1, 0.5, (1.0 * sqrt((t_1 * 0.5)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_3 = Float64(1.0 / t_2)
	t_4 = Float64(sqrt(2.0) * (t_2 ^ 2.0))
	t_5 = fma(-0.0625, Float64(sqrt(0.5) / t_4), Float64(0.1875 * t_3))
	t_6 = Float64(0.5 + Float64(t_0 * 0.5))
	tmp = 0.0
	if (x_m <= 0.0085)
		tmp = Float64((x_m ^ 2.0) * fma((x_m ^ 2.0), Float64(Float64((x_m ^ 2.0) * Float64(Float64(0.15625 * t_3) - fma(0.125, Float64(Float64(sqrt(0.5) * Float64(0.375 - Float64(0.0625 * Float64(1.0 / (sqrt(2.0) ^ 2.0))))) / t_4), Float64(0.25 * Float64(Float64(sqrt(0.5) * t_5) / Float64(sqrt(2.0) * t_2)))))) - t_5), Float64(0.25 * t_3)));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_6 ^ 4.5)) / Float64(1.0 + fma((t_1 ^ 3.0), 0.125, (t_6 ^ 1.5)))) / Float64(1.0 + fma(t_1, 0.5, Float64(1.0 * sqrt(Float64(t_1 * 0.5))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$4), $MachinePrecision] + N[(0.1875 * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0085], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.15625 * t$95$3), $MachinePrecision] - N[(0.125 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.375 - N[(0.0625 * N[(1.0 / N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(0.25 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$5), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision] + N[(0.25 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$6, 4.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$1, 3.0], $MachinePrecision] * 0.125 + N[Power[t$95$6, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * 0.5 + N[(1.0 * N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0085000000000000006

   1. Initial program 51.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(\frac{1}{8} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{1}{4} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right) - \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(0.15625 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}} - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.25 \cdot \frac{\sqrt{0.5} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}\right)\right) - \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)} \]

   if 0.0085000000000000006 < 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. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

   5. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{1 - {\left(0.5 + \cos \tan^{-1} x \cdot 0.5\right)}^{4.5}}{1 + \mathsf{fma}\left({\left(\cos \tan^{-1} x + 1\right)}^{3}, 0.125, {\left(0.5 + \cos \tan^{-1} x \cdot 0.5\right)}^{1.5}\right)}}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \frac{1}{t\_0}\\ t_2 := \cos \tan^{-1} x\_m + 1\\ t_3 := \sqrt{2} \cdot {t\_0}^{2}\\ t_4 := \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{t\_3}, 0.1875 \cdot t\_1\right)\\ t_5 := t\_2 \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0115:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left({x\_m}^{2}, {x\_m}^{2} \cdot \left(0.15625 \cdot t\_1 - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{t\_3}, 0.25 \cdot \frac{\sqrt{0.5} \cdot t\_4}{\sqrt{2} \cdot t\_0}\right)\right) - t\_4, 0.25 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_5}^{1.5}}{1 + \mathsf{fma}\left(t\_2, 0.5, 1 \cdot \sqrt{t\_5}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ (cos (atan x_m)) 1.0))
        (t_3 (* (sqrt 2.0) (pow t_0 2.0)))
        (t_4 (fma -0.0625 (/ (sqrt 0.5) t_3) (* 0.1875 t_1)))
        (t_5 (* t_2 0.5)))
   (if (<= x_m 0.0115)
     (*
      (pow x_m 2.0)
      (fma
       (pow x_m 2.0)
       (-
        (*
         (pow x_m 2.0)
         (-
          (* 0.15625 t_1)
          (fma
           0.125
           (/
            (* (sqrt 0.5) (- 0.375 (* 0.0625 (/ 1.0 (pow (sqrt 2.0) 2.0)))))
            t_3)
           (* 0.25 (/ (* (sqrt 0.5) t_4) (* (sqrt 2.0) t_0))))))
        t_4)
       (* 0.25 t_1)))
     (/ (- 1.0 (pow t_5 1.5)) (+ 1.0 (fma t_2 0.5 (* 1.0 (sqrt t_5))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = 1.0 / t_0;
	double t_2 = cos(atan(x_m)) + 1.0;
	double t_3 = sqrt(2.0) * pow(t_0, 2.0);
	double t_4 = fma(-0.0625, (sqrt(0.5) / t_3), (0.1875 * t_1));
	double t_5 = t_2 * 0.5;
	double tmp;
	if (x_m <= 0.0115) {
		tmp = pow(x_m, 2.0) * fma(pow(x_m, 2.0), ((pow(x_m, 2.0) * ((0.15625 * t_1) - fma(0.125, ((sqrt(0.5) * (0.375 - (0.0625 * (1.0 / pow(sqrt(2.0), 2.0))))) / t_3), (0.25 * ((sqrt(0.5) * t_4) / (sqrt(2.0) * t_0)))))) - t_4), (0.25 * t_1));
	} else {
		tmp = (1.0 - pow(t_5, 1.5)) / (1.0 + fma(t_2, 0.5, (1.0 * sqrt(t_5))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(cos(atan(x_m)) + 1.0)
	t_3 = Float64(sqrt(2.0) * (t_0 ^ 2.0))
	t_4 = fma(-0.0625, Float64(sqrt(0.5) / t_3), Float64(0.1875 * t_1))
	t_5 = Float64(t_2 * 0.5)
	tmp = 0.0
	if (x_m <= 0.0115)
		tmp = Float64((x_m ^ 2.0) * fma((x_m ^ 2.0), Float64(Float64((x_m ^ 2.0) * Float64(Float64(0.15625 * t_1) - fma(0.125, Float64(Float64(sqrt(0.5) * Float64(0.375 - Float64(0.0625 * Float64(1.0 / (sqrt(2.0) ^ 2.0))))) / t_3), Float64(0.25 * Float64(Float64(sqrt(0.5) * t_4) / Float64(sqrt(2.0) * t_0)))))) - t_4), Float64(0.25 * t_1)));
	else
		tmp = Float64(Float64(1.0 - (t_5 ^ 1.5)) / Float64(1.0 + fma(t_2, 0.5, Float64(1.0 * sqrt(t_5)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.1875 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0115], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.15625 * t$95$1), $MachinePrecision] - N[(0.125 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.375 - N[(0.0625 * N[(1.0 / N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.25 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$4), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$5, 1.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * 0.5 + N[(1.0 * N[Sqrt[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0115

   1. Initial program 51.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(\frac{1}{8} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{1}{4} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right) - \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(0.15625 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}} - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.25 \cdot \frac{\sqrt{0.5} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}\right)\right) - \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)} \]

   if 0.0115 < 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. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m + 1\\ t_1 := t\_0 \cdot 0.5\\ t_2 := 1 + \mathsf{fma}\left(t\_0, 0.5, 1 \cdot \sqrt{t\_1}\right)\\ \mathbf{if}\;x\_m \leq 0.0115:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.375 + {x\_m}^{2} \cdot \left(0.2685546875 \cdot {x\_m}^{2} - 0.3046875\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_1}^{1.5}}{t\_2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (cos (atan x_m)) 1.0))
        (t_1 (* t_0 0.5))
        (t_2 (+ 1.0 (fma t_0 0.5 (* 1.0 (sqrt t_1))))))
   (if (<= x_m 0.0115)
     (/
      (*
       (pow x_m 2.0)
       (+
        0.375
        (* (pow x_m 2.0) (- (* 0.2685546875 (pow x_m 2.0)) 0.3046875))))
      t_2)
     (/ (- 1.0 (pow t_1 1.5)) t_2))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m)) + 1.0;
	double t_1 = t_0 * 0.5;
	double t_2 = 1.0 + fma(t_0, 0.5, (1.0 * sqrt(t_1)));
	double tmp;
	if (x_m <= 0.0115) {
		tmp = (pow(x_m, 2.0) * (0.375 + (pow(x_m, 2.0) * ((0.2685546875 * pow(x_m, 2.0)) - 0.3046875)))) / t_2;
	} else {
		tmp = (1.0 - pow(t_1, 1.5)) / t_2;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(cos(atan(x_m)) + 1.0)
	t_1 = Float64(t_0 * 0.5)
	t_2 = Float64(1.0 + fma(t_0, 0.5, Float64(1.0 * sqrt(t_1))))
	tmp = 0.0
	if (x_m <= 0.0115)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.375 + Float64((x_m ^ 2.0) * Float64(Float64(0.2685546875 * (x_m ^ 2.0)) - 0.3046875)))) / t_2);
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 1.5)) / t_2);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(t$95$0 * 0.5 + N[(1.0 * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0115], N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.375 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.2685546875 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.3046875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0115

   1. Initial program 51.8%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 51.8%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-fma.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)\right)}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \color{blue}{\left(\frac{3}{8} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)\right)}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\color{blue}{\frac{3}{8}} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{3}{8} + \color{blue}{{x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)}\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \color{blue}{\left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)}\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \left(\color{blue}{\frac{275}{1024} \cdot {x}^{2}} - \frac{39}{128}\right)\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \color{blue}{\frac{39}{128}}\right)\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, \frac{1}{2}, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}\right)} \]

      8. lower-pow.f64 99.8

         \[\leadsto \frac{{x}^{2} \cdot \left(0.375 + {x}^{2} \cdot \left(0.2685546875 \cdot {x}^{2} - 0.3046875\right)\right)}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]

   8. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(0.375 + {x}^{2} \cdot \left(0.2685546875 \cdot {x}^{2} - 0.3046875\right)\right)}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, 1 \cdot \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} \]

   if 0.0115 < 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. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \frac{1}{t\_0}\\ t_2 := \cos \tan^{-1} x\_m + 1\\ t_3 := t\_2 \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0025:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {t\_0}^{2}}, 0.1875 \cdot t\_1\right), 0.25 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_3}^{1.5}}{1 + \mathsf{fma}\left(t\_2, 0.5, 1 \cdot \sqrt{t\_3}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ (cos (atan x_m)) 1.0))
        (t_3 (* t_2 0.5)))
   (if (<= x_m 0.0025)
     (*
      (pow x_m 2.0)
      (fma
       -1.0
       (*
        (pow x_m 2.0)
        (fma
         -0.0625
         (/ (sqrt 0.5) (* (sqrt 2.0) (pow t_0 2.0)))
         (* 0.1875 t_1)))
       (* 0.25 t_1)))
     (/ (- 1.0 (pow t_3 1.5)) (+ 1.0 (fma t_2 0.5 (* 1.0 (sqrt t_3))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = 1.0 / t_0;
	double t_2 = cos(atan(x_m)) + 1.0;
	double t_3 = t_2 * 0.5;
	double tmp;
	if (x_m <= 0.0025) {
		tmp = pow(x_m, 2.0) * fma(-1.0, (pow(x_m, 2.0) * fma(-0.0625, (sqrt(0.5) / (sqrt(2.0) * pow(t_0, 2.0))), (0.1875 * t_1))), (0.25 * t_1));
	} else {
		tmp = (1.0 - pow(t_3, 1.5)) / (1.0 + fma(t_2, 0.5, (1.0 * sqrt(t_3))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(cos(atan(x_m)) + 1.0)
	t_3 = Float64(t_2 * 0.5)
	tmp = 0.0
	if (x_m <= 0.0025)
		tmp = Float64((x_m ^ 2.0) * fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.0625, Float64(sqrt(0.5) / Float64(sqrt(2.0) * (t_0 ^ 2.0))), Float64(0.1875 * t_1))), Float64(0.25 * t_1)));
	else
		tmp = Float64(Float64(1.0 - (t_3 ^ 1.5)) / Float64(1.0 + fma(t_2, 0.5, Float64(1.0 * sqrt(t_3)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0025], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$3, 1.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$2 * 0.5 + N[(1.0 * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00250000000000000005

   1. Initial program 51.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)} + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {x}^{2} \cdot \mathsf{fma}\left(-1, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}, \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \mathsf{fma}\left(-1, {x}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)} \]

   if 0.00250000000000000005 < 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. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \cos \tan^{-1} x\_m + 1\\ t_2 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 0.0025:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {t\_0}^{2}}, 0.1875 \cdot t\_2\right), 0.25 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(t\_1, 0.5, 1 \cdot \sqrt{t\_1 \cdot 0.5}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (+ (cos (atan x_m)) 1.0))
        (t_2 (/ 1.0 t_0)))
   (if (<= x_m 0.0025)
     (*
      (pow x_m 2.0)
      (fma
       -1.0
       (*
        (pow x_m 2.0)
        (fma
         -0.0625
         (/ (sqrt 0.5) (* (sqrt 2.0) (pow t_0 2.0)))
         (* 0.1875 t_2)))
       (* 0.25 t_2)))
     (/
      (- 1.0 (pow (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5) 1.5))
      (+ 1.0 (fma t_1 0.5 (* 1.0 (sqrt (* t_1 0.5)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = cos(atan(x_m)) + 1.0;
	double t_2 = 1.0 / t_0;
	double tmp;
	if (x_m <= 0.0025) {
		tmp = pow(x_m, 2.0) * fma(-1.0, (pow(x_m, 2.0) * fma(-0.0625, (sqrt(0.5) / (sqrt(2.0) * pow(t_0, 2.0))), (0.1875 * t_2))), (0.25 * t_2));
	} else {
		tmp = (1.0 - pow(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5), 1.5)) / (1.0 + fma(t_1, 0.5, (1.0 * sqrt((t_1 * 0.5)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(cos(atan(x_m)) + 1.0)
	t_2 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 0.0025)
		tmp = Float64((x_m ^ 2.0) * fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.0625, Float64(sqrt(0.5) / Float64(sqrt(2.0) * (t_0 ^ 2.0))), Float64(0.1875 * t_2))), Float64(0.25 * t_2)));
	else
		tmp = Float64(Float64(1.0 - (Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5) ^ 1.5)) / Float64(1.0 + fma(t_1, 0.5, Float64(1.0 * sqrt(Float64(t_1 * 0.5))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0025], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * 0.5 + N[(1.0 * N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00250000000000000005

   1. Initial program 51.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)} + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {x}^{2} \cdot \mathsf{fma}\left(-1, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}, \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \mathsf{fma}\left(-1, {x}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)} \]

   if 0.00250000000000000005 < 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. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. sqrt-undiv N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 6: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \frac{1}{t\_0}\\ t_2 := \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.0024:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {t\_0}^{2}}, 0.1875 \cdot t\_1\right), 0.25 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_2}{1 + \sqrt{t\_2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (/ 1.0 t_0))
        (t_2 (* (+ (cos (atan x_m)) 1.0) 0.5)))
   (if (<= x_m 0.0024)
     (*
      (pow x_m 2.0)
      (fma
       -1.0
       (*
        (pow x_m 2.0)
        (fma
         -0.0625
         (/ (sqrt 0.5) (* (sqrt 2.0) (pow t_0 2.0)))
         (* 0.1875 t_1)))
       (* 0.25 t_1)))
     (/ (- 1.0 t_2) (+ 1.0 (sqrt t_2))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = 1.0 / t_0;
	double t_2 = (cos(atan(x_m)) + 1.0) * 0.5;
	double tmp;
	if (x_m <= 0.0024) {
		tmp = pow(x_m, 2.0) * fma(-1.0, (pow(x_m, 2.0) * fma(-0.0625, (sqrt(0.5) / (sqrt(2.0) * pow(t_0, 2.0))), (0.1875 * t_1))), (0.25 * t_1));
	} else {
		tmp = (1.0 - t_2) / (1.0 + sqrt(t_2));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.0024)
		tmp = Float64((x_m ^ 2.0) * fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.0625, Float64(sqrt(0.5) / Float64(sqrt(2.0) * (t_0 ^ 2.0))), Float64(0.1875 * t_1))), Float64(0.25 * t_1)));
	else
		tmp = Float64(Float64(1.0 - t_2) / Float64(1.0 + sqrt(t_2)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0024], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$2), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00239999999999999979

   1. Initial program 51.8%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)} + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {x}^{2} \cdot \mathsf{fma}\left(-1, \color{blue}{{x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}, \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \]

   9. Applied rewrites 99.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \mathsf{fma}\left(-1, {x}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)} \]

   if 0.00239999999999999979 < 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. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.000105:\\ \;\;\;\;0.25 \cdot \frac{{x\_m}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (+ (cos (atan x_m)) 1.0) 0.5)))
   (if (<= x_m 0.000105)
     (* 0.25 (/ (pow x_m 2.0) (+ 1.0 (* (sqrt 0.5) (sqrt 2.0)))))
     (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (cos(atan(x_m)) + 1.0) * 0.5;
	double tmp;
	if (x_m <= 0.000105) {
		tmp = 0.25 * (pow(x_m, 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cos(atan(x_m)) + 1.0d0) * 0.5d0
    if (x_m <= 0.000105d0) then
        tmp = 0.25d0 * ((x_m ** 2.0d0) / (1.0d0 + (sqrt(0.5d0) * sqrt(2.0d0))))
    else
        tmp = (1.0d0 - t_0) / (1.0d0 + sqrt(t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (Math.cos(Math.atan(x_m)) + 1.0) * 0.5;
	double tmp;
	if (x_m <= 0.000105) {
		tmp = 0.25 * (Math.pow(x_m, 2.0) / (1.0 + (Math.sqrt(0.5) * Math.sqrt(2.0))));
	} else {
		tmp = (1.0 - t_0) / (1.0 + Math.sqrt(t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (math.cos(math.atan(x_m)) + 1.0) * 0.5
	tmp = 0
	if x_m <= 0.000105:
		tmp = 0.25 * (math.pow(x_m, 2.0) / (1.0 + (math.sqrt(0.5) * math.sqrt(2.0))))
	else:
		tmp = (1.0 - t_0) / (1.0 + math.sqrt(t_0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.000105)
		tmp = Float64(0.25 * Float64((x_m ^ 2.0) / Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (cos(atan(x_m)) + 1.0) * 0.5;
	tmp = 0.0;
	if (x_m <= 0.000105)
		tmp = 0.25 * ((x_m ^ 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	else
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.000105], N[(0.25 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] / N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e-4

   1. Initial program 51.7%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{\color{blue}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{\color{blue}{1} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}} \]

      7. lower-sqrt.f64 99.7

         \[\leadsto 0.25 \cdot \frac{{x}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}} \]

   9. Applied rewrites 99.7%

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

   if 1.05e-4 < x

   1. Initial program 98.2%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ \mathbf{if}\;x\_m \leq 9.2 \cdot 10^{-5}:\\ \;\;\;\;0.25 \cdot \frac{{x\_m}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - 0.5 \cdot t\_0}{1 + \sqrt{0.5 + t\_0 \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m))))
   (if (<= x_m 9.2e-5)
     (* 0.25 (/ (pow x_m 2.0) (+ 1.0 (* (sqrt 0.5) (sqrt 2.0)))))
     (/ (- 0.5 (* 0.5 t_0)) (+ 1.0 (sqrt (+ 0.5 (* t_0 0.5))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double tmp;
	if (x_m <= 9.2e-5) {
		tmp = 0.25 * (pow(x_m, 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	} else {
		tmp = (0.5 - (0.5 * t_0)) / (1.0 + sqrt((0.5 + (t_0 * 0.5))));
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(atan(x_m))
    if (x_m <= 9.2d-5) then
        tmp = 0.25d0 * ((x_m ** 2.0d0) / (1.0d0 + (sqrt(0.5d0) * sqrt(2.0d0))))
    else
        tmp = (0.5d0 - (0.5d0 * t_0)) / (1.0d0 + sqrt((0.5d0 + (t_0 * 0.5d0))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = Math.cos(Math.atan(x_m));
	double tmp;
	if (x_m <= 9.2e-5) {
		tmp = 0.25 * (Math.pow(x_m, 2.0) / (1.0 + (Math.sqrt(0.5) * Math.sqrt(2.0))));
	} else {
		tmp = (0.5 - (0.5 * t_0)) / (1.0 + Math.sqrt((0.5 + (t_0 * 0.5))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = math.cos(math.atan(x_m))
	tmp = 0
	if x_m <= 9.2e-5:
		tmp = 0.25 * (math.pow(x_m, 2.0) / (1.0 + (math.sqrt(0.5) * math.sqrt(2.0))))
	else:
		tmp = (0.5 - (0.5 * t_0)) / (1.0 + math.sqrt((0.5 + (t_0 * 0.5))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	tmp = 0.0
	if (x_m <= 9.2e-5)
		tmp = Float64(0.25 * Float64((x_m ^ 2.0) / Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))));
	else
		tmp = Float64(Float64(0.5 - Float64(0.5 * t_0)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(t_0 * 0.5)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = cos(atan(x_m));
	tmp = 0.0;
	if (x_m <= 9.2e-5)
		tmp = 0.25 * ((x_m ^ 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	else
		tmp = (0.5 - (0.5 * t_0)) / (1.0 + sqrt((0.5 + (t_0 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 9.2e-5], N[(0.25 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] / N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.20000000000000001e-5

   1. Initial program 51.7%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{\color{blue}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{\color{blue}{1} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}} \]

      7. lower-sqrt.f64 99.7

         \[\leadsto 0.25 \cdot \frac{{x}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}} \]

   9. Applied rewrites 99.7%

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

   if 9.20000000000000001e-5 < x

   1. Initial program 98.2%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \tan^{-1} x}}{1 + \sqrt{\frac{1}{2} + \cos \tan^{-1} x \cdot \frac{1}{2}}} \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \tan^{-1} x}}{1 + \sqrt{\frac{1}{2} + \cos \tan^{-1} x \cdot \frac{1}{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \tan^{-1} x}}{1 + \sqrt{\frac{1}{2} + \cos \tan^{-1} x \cdot \frac{1}{2}}} \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\frac{1}{2} - \frac{1}{2} \cdot \cos \tan^{-1} x}{1 + \sqrt{\frac{1}{2} + \cos \tan^{-1} x \cdot \frac{1}{2}}} \]

      4. lift-atan.f64 98.3

         \[\leadsto \frac{0.5 - 0.5 \cdot \cos \tan^{-1} x}{1 + \sqrt{0.5 + \cos \tan^{-1} x \cdot 0.5}} \]

   8. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{0.5 - 0.5 \cdot \cos \tan^{-1} x}}{1 + \sqrt{0.5 + \cos \tan^{-1} x \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0:\\ \;\;\;\;\frac{1 - \left(0.5 - \frac{0.5}{x\_m}\right)}{1 + \left(\sqrt{0.5} + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.125, \sqrt{0.5}, 0.1875 \cdot \frac{\sqrt{0.5}}{x\_m}\right)}{x\_m}, 0.5 \cdot \sqrt{0.5}\right)}{x\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m)))))) 0.0)
   (/
    (- 1.0 (- 0.5 (/ 0.5 x_m)))
    (+
     1.0
     (+
      (sqrt 0.5)
      (*
       -1.0
       (/
        (fma
         -1.0
         (/ (fma -0.125 (sqrt 0.5) (* 0.1875 (/ (sqrt 0.5) x_m))) x_m)
         (* 0.5 (sqrt 0.5)))
        x_m)))))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))))) <= 0.0) {
		tmp = (1.0 - (0.5 - (0.5 / x_m))) / (1.0 + (sqrt(0.5) + (-1.0 * (fma(-1.0, (fma(-0.125, sqrt(0.5), (0.1875 * (sqrt(0.5) / x_m))) / x_m), (0.5 * sqrt(0.5))) / x_m))));
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m)))))) <= 0.0)
		tmp = Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 / x_m))) / Float64(1.0 + Float64(sqrt(0.5) + Float64(-1.0 * Float64(fma(-1.0, Float64(fma(-0.125, sqrt(0.5), Float64(0.1875 * Float64(sqrt(0.5) / x_m))) / x_m), Float64(0.5 * sqrt(0.5))) / x_m)))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(1.0 - N[(0.5 - N[(0.5 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] + N[(-1.0 * N[(N[(-1.0 * N[(N[(-0.125 * N[Sqrt[0.5], $MachinePrecision] + N[(0.1875 * N[(N[Sqrt[0.5], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(0.5 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))) < 0.0

   1. Initial program 51.9%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \frac{1 - \left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right)}{\color{blue}{1 + \left(\sqrt{\frac{1}{2}} + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{2}} + \frac{3}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{x}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{x}\right)}} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right)}{1 + \color{blue}{\left(\sqrt{\frac{1}{2}} + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{2}} + \frac{3}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{x}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{x}\right)}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right)}{1 + \left(\sqrt{\frac{1}{2}} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{2}} + \frac{3}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{x}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{x}}\right)} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right)}{1 + \left(\sqrt{\frac{1}{2}} + \color{blue}{-1} \cdot \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{2}} + \frac{3}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{x}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{x}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right)}{1 + \left(\sqrt{\frac{1}{2}} + -1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{2}} + \frac{3}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{x}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{x}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{2} - \frac{\frac{1}{2}}{x}\right)}{1 + \left(\sqrt{\frac{1}{2}} + -1 \cdot \frac{-1 \cdot \frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{2}} + \frac{3}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{x}}{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{2}}}{\color{blue}{x}}\right)} \]

   9. Applied rewrites 55.6%

      \[\leadsto \frac{1 - \left(0.5 - \frac{0.5}{x}\right)}{\color{blue}{1 + \left(\sqrt{0.5} + -1 \cdot \frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-0.125, \sqrt{0.5}, 0.1875 \cdot \frac{\sqrt{0.5}}{x}\right)}{x}, 0.5 \cdot \sqrt{0.5}\right)}{x}\right)}} \]

   if 0.0 < (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))))

   1. Initial program 97.4%

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

      1. lift-hypot.f64 N/A

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 97.4

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

   3. Applied rewrites 97.4%

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

Alternative 10: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.04:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x\_m \cdot x\_m\right) - 0.25, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m)))))) 0.04)
   (- 1.0 (sqrt (fma (- (* 0.1875 (* x_m x_m)) 0.25) (* x_m x_m) 1.0)))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))))) <= 0.04) {
		tmp = 1.0 - sqrt(fma(((0.1875 * (x_m * x_m)) - 0.25), (x_m * x_m), 1.0));
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m)))))) <= 0.04)
		tmp = Float64(1.0 - sqrt(fma(Float64(Float64(0.1875 * Float64(x_m * x_m)) - 0.25), Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.04], N[(1.0 - N[Sqrt[N[(N[(N[(0.1875 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))) < 0.0400000000000000008

   1. Initial program 52.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \sqrt{\color{blue}{1 + {x}^{2} \cdot \left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto 1 - \sqrt{{x}^{2} \cdot \left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}} \]

      2. *-commutative N/A

         \[\leadsto 1 - \sqrt{\left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)} \]

      4. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, {\color{blue}{x}}^{2}, 1\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\frac{3}{16} \cdot {x}^{2} - \frac{1}{4}, {x}^{2}, 1\right)} \]

      6. pow2 N/A

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\frac{3}{16} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\frac{3}{16} \cdot \left(x \cdot x\right) - \frac{1}{4}, {x}^{2}, 1\right)} \]

      8. pow2 N/A

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(\frac{3}{16} \cdot \left(x \cdot x\right) - \frac{1}{4}, x \cdot \color{blue}{x}, 1\right)} \]

      9. lower-*.f64 51.7

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 51.7%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   if 0.0400000000000000008 < (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))))

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}} \]

      5. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 11: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.125, x\_m \cdot x\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))
   (- 1.0 (fma -0.125 (* x_m x_m) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	} else {
		tmp = 1.0 - fma(-0.125, (x_m * x_m), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	else
		tmp = Float64(1.0 - fma(-0.125, Float64(x_m * x_m), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(-0.125 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2} \cdot 1}{x} + \frac{1}{2}} \]

      4. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{\frac{1}{2}}{x} + \frac{1}{2}} \]

      5. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 52.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{-1}{4} \cdot \frac{{x}^{2} \cdot \sqrt{\frac{1}{2}}}{\sqrt{2}} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \left(\frac{{x}^{2} \cdot \sqrt{\frac{1}{2}}}{\sqrt{2}} \cdot \frac{-1}{4} + \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right) \cdot \frac{-1}{4} + \sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      3. sqrt-undiv N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \sqrt{\frac{\frac{1}{2}}{2}}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      4. metadata-eval N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \sqrt{\frac{1}{4}}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      6. *-commutative N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2} \cdot 2}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{2} \cdot {x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right) \]

      11. *-commutative N/A

          \[\leadsto 1 - \mathsf{fma}\left({x}^{2} \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 1 - \mathsf{fma}\left({x}^{2} \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      13. pow2 N/A

          \[\leadsto 1 - \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      14. lower-*.f64 51.5

          \[\leadsto 1 - \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, -0.25, 1\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{1 - \mathsf{fma}\left(-0.125, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.8:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.125, x\_m \cdot x\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.8)
   (/ 0.5 (+ 1.0 (sqrt 0.5)))
   (- 1.0 (fma -0.125 (* x_m x_m) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m))))) <= 0.8) {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	} else {
		tmp = 1.0 - fma(-0.125, (x_m * x_m), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))) <= 0.8)
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	else
		tmp = Float64(1.0 - fma(-0.125, Float64(x_m * x_m), 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(-0.125 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   1. Initial program 98.5%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 96.5

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

   4. Applied rewrites 96.5%

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

      1. metadata-eval 96.5

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{\color{blue}{1 + \sqrt{\frac{1}{2}}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{2}}{1 + \color{blue}{\sqrt{\frac{1}{2}}}} \]

      3. lower-sqrt.f64 98.0

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

   9. Applied rewrites 98.0%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 52.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{-1}{4} \cdot \frac{{x}^{2} \cdot \sqrt{\frac{1}{2}}}{\sqrt{2}} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \left(\frac{{x}^{2} \cdot \sqrt{\frac{1}{2}}}{\sqrt{2}} \cdot \frac{-1}{4} + \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right) \cdot \frac{-1}{4} + \sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      3. sqrt-undiv N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \sqrt{\frac{\frac{1}{2}}{2}}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      4. metadata-eval N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \sqrt{\frac{1}{4}}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      6. *-commutative N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2} \cdot 2}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{2} \cdot {x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right) \]

      11. *-commutative N/A

          \[\leadsto 1 - \mathsf{fma}\left({x}^{2} \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 1 - \mathsf{fma}\left({x}^{2} \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      13. pow2 N/A

          \[\leadsto 1 - \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      14. lower-*.f64 51.5

          \[\leadsto 1 - \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, -0.25, 1\right) \]

   4. Applied rewrites 51.5%

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

   6. Applied rewrites 51.5%

      \[\leadsto \color{blue}{1 - \mathsf{fma}\left(-0.125, x \cdot x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.04:\\ \;\;\;\;1 - \mathsf{fma}\left(-0.125, x\_m \cdot x\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m)))))) 0.04)
   (- 1.0 (fma -0.125 (* x_m x_m) 1.0))
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))))) <= 0.04) {
		tmp = 1.0 - fma(-0.125, (x_m * x_m), 1.0);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m)))))) <= 0.04)
		tmp = Float64(1.0 - fma(-0.125, Float64(x_m * x_m), 1.0));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.04], N[(1.0 - N[(-0.125 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))) < 0.0400000000000000008

   1. Initial program 52.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto 1 - \color{blue}{\left(\frac{-1}{4} \cdot \frac{{x}^{2} \cdot \sqrt{\frac{1}{2}}}{\sqrt{2}} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto 1 - \left(\frac{{x}^{2} \cdot \sqrt{\frac{1}{2}}}{\sqrt{2}} \cdot \frac{-1}{4} + \color{blue}{\sqrt{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      2. associate-/l* N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right) \cdot \frac{-1}{4} + \sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      3. sqrt-undiv N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \sqrt{\frac{\frac{1}{2}}{2}}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      4. metadata-eval N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \sqrt{\frac{1}{4}}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto 1 - \left(\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right) \]

      6. *-commutative N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{\color{blue}{\frac{1}{2}}} \cdot \sqrt{2}\right) \]

      7. sqrt-unprod N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{\frac{1}{2} \cdot 2}\right) \]

      8. metadata-eval N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + \sqrt{1}\right) \]

      9. metadata-eval N/A

         \[\leadsto 1 - \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{-1}{4} + 1\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto 1 - \mathsf{fma}\left(\frac{1}{2} \cdot {x}^{2}, \color{blue}{\frac{-1}{4}}, 1\right) \]

      11. *-commutative N/A

          \[\leadsto 1 - \mathsf{fma}\left({x}^{2} \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      12. lower-*.f64 N/A

          \[\leadsto 1 - \mathsf{fma}\left({x}^{2} \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      13. pow2 N/A

          \[\leadsto 1 - \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{2}, \frac{-1}{4}, 1\right) \]

      14. lower-*.f64 51.6

          \[\leadsto 1 - \mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, -0.25, 1\right) \]

   4. Applied rewrites 51.6%

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

   6. Applied rewrites 51.6%

      \[\leadsto \color{blue}{1 - \mathsf{fma}\left(-0.125, x \cdot x, 1\right)} \]

   if 0.0400000000000000008 < (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))))

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 96.4%

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

Alternative 14: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000112:\\ \;\;\;\;0.25 \cdot \frac{{x\_m}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000112)
   (* 0.25 (/ (pow x_m 2.0) (+ 1.0 (* (sqrt 0.5) (sqrt 2.0)))))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000112) {
		tmp = 0.25 * (pow(x_m, 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000112)
		tmp = Float64(0.25 * Float64((x_m ^ 2.0) / Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000112], N[(0.25 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] / N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.11999999999999998e-4

   1. Initial program 51.7%

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

   2. Taylor expanded in x around -inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \frac{1}{x}}} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \frac{1}{x}}} \]

      2. associate-*r/ N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2} \cdot 1}{\color{blue}{x}}} \]

      3. metadata-eval N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}} \]

      4. lower-/.f64 0.0

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

   4. Applied rewrites 0.0%

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

      1. metadata-eval 0.0

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

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2} - \frac{\frac{1}{2}}{x}}} \]

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \color{blue}{\frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{\color{blue}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{\color{blue}{1} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \color{blue}{\sqrt{2}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{2}}} \]

      7. lower-sqrt.f64 99.7

         \[\leadsto 0.25 \cdot \frac{{x}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}} \]

   9. Applied rewrites 99.7%

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

   if 1.11999999999999998e-4 < x

   1. Initial program 98.2%

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

      1. lift-hypot.f64 N/A

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 98.2

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

   3. Applied rewrites 98.2%

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

Alternative 15: 74.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. lift-hypot.f64 N/A

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

   2. metadata-eval N/A

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

   3. lower-sqrt.f64 N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. pow2 N/A

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

   7. lower-fma.f64 74.9

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

3. Applied rewrites 74.9%

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

Alternative 16: 73.6% accurate, 6.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.2e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

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

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.2e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   1. Initial program 67.7%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   3. Step-by-step derivation

      1. sqrt-unprod N/A

         \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot 2} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \sqrt{1} \]

      3. metadata-eval N/A

         \[\leadsto 1 - 1 \]

      4. metadata-eval 67.7

         \[\leadsto 0 \]

   4. Applied rewrites 67.7%

      \[\leadsto \color{blue}{0} \]

   if 2.20000000000000007e-77 < x

   1. Initial program 79.1%

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

   2. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 77.2%

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

Alternative 17: 27.3% accurate, 134.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 74.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
3. Step-by-step derivation

   1. sqrt-unprod N/A

      \[\leadsto 1 - \sqrt{\frac{1}{2} \cdot 2} \]

   2. metadata-eval N/A

      \[\leadsto 1 - \sqrt{1} \]

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 27.3

      \[\leadsto 0 \]

4. Applied rewrites 27.3%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))