Given's Rotation SVD example, simplified

Percentage Accurate: 75.6% → 99.9%

Time: 5.2s

Alternatives: 14

Speedup: 6.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.6% 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, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5\\ t_1 := \sqrt{t\_0} + 1\\ t_2 := {t\_1}^{-1}\\ t_3 := \frac{t\_0}{t\_1}\\ t_4 := \mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)\\ t_5 := 0.07877604166666667 - \frac{t\_4}{9} \cdot 0.375\\ \mathbf{if}\;x\_m \leq 0.0275:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(t\_4 \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{t\_5}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), t\_5\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_2}^{3} - {t\_3}^{3}}{\mathsf{fma}\left(t\_2, t\_2, \mathsf{fma}\left(t\_3, t\_3, t\_2 \cdot t\_3\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (+ (cos (atan x_m)) 1.0) 0.5))
        (t_1 (+ (sqrt t_0) 1.0))
        (t_2 (pow t_1 -1.0))
        (t_3 (/ t_0 t_1))
        (t_4 (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875))
        (t_5 (- 0.07877604166666667 (* (/ t_4 9.0) 0.375))))
   (if (<= x_m 0.0275)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (fma (* t_4 0.028645833333333332) -1.0 0.081817626953125)
           (fma
            -0.375
            (/ t_5 3.0)
            (*
             (/
              (- (/ (* -0.5 (* 0.26953125 (sqrt 0.5))) (sqrt 2.0)) 0.15625)
              9.0)
             0.375)))
          t_5)
         (* x_m x_m))
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/
      (- (pow t_2 3.0) (pow t_3 3.0))
      (fma t_2 t_2 (fma t_3 t_3 (* t_2 t_3)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (cos(atan(x_m)) + 1.0) * 0.5;
	double t_1 = sqrt(t_0) + 1.0;
	double t_2 = pow(t_1, -1.0);
	double t_3 = t_0 / t_1;
	double t_4 = fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875);
	double t_5 = 0.07877604166666667 - ((t_4 / 9.0) * 0.375);
	double tmp;
	if (x_m <= 0.0275) {
		tmp = fma(((fma((-x_m * x_m), (fma((t_4 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, (t_5 / 3.0), (((((-0.5 * (0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_5) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (pow(t_2, 3.0) - pow(t_3, 3.0)) / fma(t_2, t_2, fma(t_3, t_3, (t_2 * t_3)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)
	t_1 = Float64(sqrt(t_0) + 1.0)
	t_2 = t_1 ^ -1.0
	t_3 = Float64(t_0 / t_1)
	t_4 = fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875)
	t_5 = Float64(0.07877604166666667 - Float64(Float64(t_4 / 9.0) * 0.375))
	tmp = 0.0
	if (x_m <= 0.0275)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(fma(Float64(t_4 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, Float64(t_5 / 3.0), Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_5) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64((t_2 ^ 3.0) - (t_3 ^ 3.0)) / fma(t_2, t_2, fma(t_3, t_3, Float64(t_2 * t_3))));
	end
	return 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]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -1.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision]}, Block[{t$95$5 = N[(0.07877604166666667 - N[(N[(t$95$4 / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0275], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(t$95$4 * 0.028645833333333332), $MachinePrecision] * -1.0 + 0.081817626953125), $MachinePrecision] + N[(-0.375 * N[(t$95$5 / 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5 * N[(0.26953125 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] - N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(t$95$3 * t$95$3 + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0275000000000000001

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \frac{\left(\frac{3}{16} + \frac{1}{2} \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}}\right) \cdot \left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{8043}{32768} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{3}{8} \cdot \frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{16} + \frac{-1}{4} \cdot \frac{\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} - \frac{5}{32}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{\left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right) \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0275000000000000001 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. metadata-eval N/A

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

      10. rem-square-sqrt N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0275:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1\right)}^{-1}\right)}^{3} - {\left(\frac{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1}\right)}^{3}}{\mathsf{fma}\left({\left(\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1\right)}^{-1}, {\left(\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1\right)}^{-1}, \mathsf{fma}\left(\frac{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1}, \frac{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1}, {\left(\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1\right)}^{-1} \cdot \frac{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{\sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5} + 1}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m + 1\\ t_1 := \mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)\\ t_2 := 0.07877604166666667 - \frac{t\_1}{9} \cdot 0.375\\ \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(t\_1 \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{t\_2}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), t\_2\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0 \cdot 0.5}{\mathsf{fma}\left(\sqrt{t\_0}, {\left(\sqrt{2}\right)}^{-1}, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (cos (atan x_m)) 1.0))
        (t_1 (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875))
        (t_2 (- 0.07877604166666667 (* (/ t_1 9.0) 0.375))))
   (if (<= x_m 0.029)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (fma (* t_1 0.028645833333333332) -1.0 0.081817626953125)
           (fma
            -0.375
            (/ t_2 3.0)
            (*
             (/
              (- (/ (* -0.5 (* 0.26953125 (sqrt 0.5))) (sqrt 2.0)) 0.15625)
              9.0)
             0.375)))
          t_2)
         (* x_m x_m))
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/ (- 1.0 (* t_0 0.5)) (fma (sqrt t_0) (pow (sqrt 2.0) -1.0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m)) + 1.0;
	double t_1 = fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875);
	double t_2 = 0.07877604166666667 - ((t_1 / 9.0) * 0.375);
	double tmp;
	if (x_m <= 0.029) {
		tmp = fma(((fma((-x_m * x_m), (fma((t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, (t_2 / 3.0), (((((-0.5 * (0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - (t_0 * 0.5)) / fma(sqrt(t_0), pow(sqrt(2.0), -1.0), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(cos(atan(x_m)) + 1.0)
	t_1 = fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875)
	t_2 = Float64(0.07877604166666667 - Float64(Float64(t_1 / 9.0) * 0.375))
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(fma(Float64(t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, Float64(t_2 / 3.0), Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 * 0.5)) / fma(sqrt(t_0), (sqrt(2.0) ^ -1.0), 1.0));
	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[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision]}, Block[{t$95$2 = N[(0.07877604166666667 - N[(N[(t$95$1 / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.029], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.028645833333333332), $MachinePrecision] * -1.0 + 0.081817626953125), $MachinePrecision] + N[(-0.375 * N[(t$95$2 / 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5 * N[(0.26953125 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Power[N[Sqrt[2.0], $MachinePrecision], -1.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \frac{\left(\frac{3}{16} + \frac{1}{2} \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}}\right) \cdot \left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{8043}{32768} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{3}{8} \cdot \frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{16} + \frac{-1}{4} \cdot \frac{\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} - \frac{5}{32}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{\left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right) \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0290000000000000015 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. metadata-eval N/A

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

      10. rem-square-sqrt N/A

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. pow1/2 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-sqrt.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{\mathsf{fma}\left(\sqrt{\cos \tan^{-1} x + 1}, {\left(\sqrt{2}\right)}^{-1}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5\\ t_1 := \mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)\\ t_2 := 0.07877604166666667 - \frac{t\_1}{9} \cdot 0.375\\ \mathbf{if}\;x\_m \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(t\_1 \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{t\_2}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), t\_2\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}} - \frac{t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (+ (cos (atan x_m)) 1.0) 0.5))
        (t_1 (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875))
        (t_2 (- 0.07877604166666667 (* (/ t_1 9.0) 0.375))))
   (if (<= x_m 0.028)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (fma (* t_1 0.028645833333333332) -1.0 0.081817626953125)
           (fma
            -0.375
            (/ t_2 3.0)
            (*
             (/
              (- (/ (* -0.5 (* 0.26953125 (sqrt 0.5))) (sqrt 2.0)) 0.15625)
              9.0)
             0.375)))
          t_2)
         (* x_m x_m))
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (-
      (/ 1.0 (+ 1.0 (sqrt (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5))))
      (/ 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 t_1 = fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875);
	double t_2 = 0.07877604166666667 - ((t_1 / 9.0) * 0.375);
	double tmp;
	if (x_m <= 0.028) {
		tmp = fma(((fma((-x_m * x_m), (fma((t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, (t_2 / 3.0), (((((-0.5 * (0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 / (1.0 + sqrt(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)))) - (t_0 / (1.0 + 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)
	t_1 = fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875)
	t_2 = Float64(0.07877604166666667 - Float64(Float64(t_1 / 9.0) * 0.375))
	tmp = 0.0
	if (x_m <= 0.028)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(fma(Float64(t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, Float64(t_2 / 3.0), Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + sqrt(Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)))) - Float64(t_0 / Float64(1.0 + sqrt(t_0))));
	end
	return 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]}, Block[{t$95$1 = N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision]}, Block[{t$95$2 = N[(0.07877604166666667 - N[(N[(t$95$1 / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.028], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.028645833333333332), $MachinePrecision] * -1.0 + 0.081817626953125), $MachinePrecision] + N[(-0.375 * N[(t$95$2 / 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5 * N[(0.26953125 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Sqrt[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]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0280000000000000006

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \frac{\left(\frac{3}{16} + \frac{1}{2} \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}}\right) \cdot \left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{8043}{32768} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{3}{8} \cdot \frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{16} + \frac{-1}{4} \cdot \frac{\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} - \frac{5}{32}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{\left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right) \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0280000000000000006 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. metadata-eval N/A

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

      10. rem-square-sqrt N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.028:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}} + 1\right) \cdot 0.5}} - \frac{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m + 1\\ t_1 := \mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)\\ t_2 := 0.07877604166666667 - \frac{t\_1}{9} \cdot 0.375\\ \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(t\_1 \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{t\_2}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), t\_2\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0 \cdot 0.5}{\mathsf{fma}\left(\sqrt{t\_0}, \sqrt{0.5}, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (cos (atan x_m)) 1.0))
        (t_1 (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875))
        (t_2 (- 0.07877604166666667 (* (/ t_1 9.0) 0.375))))
   (if (<= x_m 0.029)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (fma (* t_1 0.028645833333333332) -1.0 0.081817626953125)
           (fma
            -0.375
            (/ t_2 3.0)
            (*
             (/
              (- (/ (* -0.5 (* 0.26953125 (sqrt 0.5))) (sqrt 2.0)) 0.15625)
              9.0)
             0.375)))
          t_2)
         (* x_m x_m))
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/ (- 1.0 (* t_0 0.5)) (fma (sqrt t_0) (sqrt 0.5) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m)) + 1.0;
	double t_1 = fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875);
	double t_2 = 0.07877604166666667 - ((t_1 / 9.0) * 0.375);
	double tmp;
	if (x_m <= 0.029) {
		tmp = fma(((fma((-x_m * x_m), (fma((t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, (t_2 / 3.0), (((((-0.5 * (0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - (t_0 * 0.5)) / fma(sqrt(t_0), sqrt(0.5), 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(cos(atan(x_m)) + 1.0)
	t_1 = fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875)
	t_2 = Float64(0.07877604166666667 - Float64(Float64(t_1 / 9.0) * 0.375))
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(fma(Float64(t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, Float64(t_2 / 3.0), Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 * 0.5)) / fma(sqrt(t_0), sqrt(0.5), 1.0));
	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[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision]}, Block[{t$95$2 = N[(0.07877604166666667 - N[(N[(t$95$1 / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.029], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.028645833333333332), $MachinePrecision] * -1.0 + 0.081817626953125), $MachinePrecision] + N[(-0.375 * N[(t$95$2 / 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5 * N[(0.26953125 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \frac{\left(\frac{3}{16} + \frac{1}{2} \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}}\right) \cdot \left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{8043}{32768} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{3}{8} \cdot \frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{16} + \frac{-1}{4} \cdot \frac{\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} - \frac{5}{32}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{\left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right) \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0290000000000000015 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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. metadata-eval N/A

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

      10. rem-square-sqrt N/A

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{\mathsf{fma}\left(\sqrt{\cos \tan^{-1} x + 1}, \sqrt{0.5}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5\\ t_1 := \mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)\\ t_2 := 0.07877604166666667 - \frac{t\_1}{9} \cdot 0.375\\ \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(t\_1 \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{t\_2}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), t\_2\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (* (+ (cos (atan x_m)) 1.0) 0.5))
        (t_1 (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875))
        (t_2 (- 0.07877604166666667 (* (/ t_1 9.0) 0.375))))
   (if (<= x_m 0.029)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (fma (* t_1 0.028645833333333332) -1.0 0.081817626953125)
           (fma
            -0.375
            (/ t_2 3.0)
            (*
             (/
              (- (/ (* -0.5 (* 0.26953125 (sqrt 0.5))) (sqrt 2.0)) 0.15625)
              9.0)
             0.375)))
          t_2)
         (* x_m x_m))
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/ (- 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 t_1 = fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875);
	double t_2 = 0.07877604166666667 - ((t_1 / 9.0) * 0.375);
	double tmp;
	if (x_m <= 0.029) {
		tmp = fma(((fma((-x_m * x_m), (fma((t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, (t_2 / 3.0), (((((-0.5 * (0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - t_0) / (1.0 + 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)
	t_1 = fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875)
	t_2 = Float64(0.07877604166666667 - Float64(Float64(t_1 / 9.0) * 0.375))
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(fma(Float64(t_1 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, Float64(t_2 / 3.0), Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_2) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return 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]}, Block[{t$95$1 = N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision]}, Block[{t$95$2 = N[(0.07877604166666667 - N[(N[(t$95$1 / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.029], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(t$95$1 * 0.028645833333333332), $MachinePrecision] * -1.0 + 0.081817626953125), $MachinePrecision] + N[(-0.375 * N[(t$95$2 / 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5 * N[(0.26953125 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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 < 0.0290000000000000015

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \frac{\left(\frac{3}{16} + \frac{1}{2} \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}}\right) \cdot \left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{8043}{32768} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{3}{8} \cdot \frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{16} + \frac{-1}{4} \cdot \frac{\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} - \frac{5}{32}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{\left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right) \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.0290000000000000015 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. 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. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)\\ t_1 := 0.07877604166666667 - \frac{t\_0}{9} \cdot 0.375\\ \mathbf{if}\;x\_m \leq 0.03:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(t\_0 \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{t\_1}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), t\_1\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875))
        (t_1 (- 0.07877604166666667 (* (/ t_0 9.0) 0.375))))
   (if (<= x_m 0.03)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (fma (* t_0 0.028645833333333332) -1.0 0.081817626953125)
           (fma
            -0.375
            (/ t_1 3.0)
            (*
             (/
              (- (/ (* -0.5 (* 0.26953125 (sqrt 0.5))) (sqrt 2.0)) 0.15625)
              9.0)
             0.375)))
          t_1)
         (* x_m x_m))
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m 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 t_0 = fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875);
	double t_1 = 0.07877604166666667 - ((t_0 / 9.0) * 0.375);
	double tmp;
	if (x_m <= 0.03) {
		tmp = fma(((fma((-x_m * x_m), (fma((t_0 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, (t_1 / 3.0), (((((-0.5 * (0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_1) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * 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)
	t_0 = fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875)
	t_1 = Float64(0.07877604166666667 - Float64(Float64(t_0 / 9.0) * 0.375))
	tmp = 0.0
	if (x_m <= 0.03)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(fma(Float64(t_0 * 0.028645833333333332), -1.0, 0.081817626953125) + fma(-0.375, Float64(t_1 / 3.0), Float64(Float64(Float64(Float64(Float64(-0.5 * Float64(0.26953125 * sqrt(0.5))) / sqrt(2.0)) - 0.15625) / 9.0) * 0.375))), t_1) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * 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_] := Block[{t$95$0 = N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision]}, Block[{t$95$1 = N[(0.07877604166666667 - N[(N[(t$95$0 / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.03], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(t$95$0 * 0.028645833333333332), $MachinePrecision] * -1.0 + 0.081817626953125), $MachinePrecision] + N[(-0.375 * N[(t$95$1 / 3.0), $MachinePrecision] + N[(N[(N[(N[(N[(-0.5 * N[(0.26953125 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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 < 0.029999999999999999

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(-1 \cdot \frac{\left(\frac{3}{16} + \frac{1}{2} \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}}\right) \cdot \left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{8043}{32768} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \left(\frac{3}{8} \cdot \frac{\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{16} + \frac{-1}{4} \cdot \frac{\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}} - \frac{5}{32}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{\left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right) \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \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}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 63.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1 \cdot \left(x \cdot x\right), \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.029999999999999999 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right) \cdot 0.028645833333333332, -1, 0.081817626953125\right) + \mathsf{fma}\left(-0.375, \frac{0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375}{3}, \frac{\frac{-0.5 \cdot \left(0.26953125 \cdot \sqrt{0.5}\right)}{\sqrt{2}} - 0.15625}{9} \cdot 0.375\right), 0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.9:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0859375 \cdot \left(x\_m \cdot x\_m\right), -1, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.9)
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))
   (* (fma (* 0.0859375 (* x_m x_m)) -1.0 0.125) (* x_m x_m))))

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))))) <= 0.9) {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	} else {
		tmp = fma((0.0859375 * (x_m * x_m)), -1.0, 0.125) * (x_m * x_m);
	}
	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))))) <= 0.9)
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	else
		tmp = Float64(fma(Float64(0.0859375 * Float64(x_m * x_m)), -1.0, 0.125) * Float64(x_m * x_m));
	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], 0.9], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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))))) < 0.900000000000000022

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 96.3

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

   5. Applied rewrites 96.3%

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

   if 0.900000000000000022 < (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 49.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 1.0

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

   5. Applied rewrites 1.0%

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

      1. metadata-eval 1.0

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

      2. cos-atan-rev 1.0

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

      3. cos-atan-rev 1.0

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 0.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.0859375 \cdot \left(x \cdot x\right), -1, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 0.9:\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0859375 \cdot \left(x\_m \cdot x\_m\right), -1, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))) 0.9)
   (- 1.0 (sqrt 0.5))
   (* (fma (* 0.0859375 (* x_m x_m)) -1.0 0.125) (* x_m x_m))))

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))))) <= 0.9) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = fma((0.0859375 * (x_m * x_m)), -1.0, 0.125) * (x_m * x_m);
	}
	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))))) <= 0.9)
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(fma(Float64(0.0859375 * Float64(x_m * x_m)), -1.0, 0.125) * Float64(x_m * x_m));
	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], 0.9], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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))))) < 0.900000000000000022

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.4%

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

   if 0.900000000000000022 < (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 49.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 1.0

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

   5. Applied rewrites 1.0%

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

      1. metadata-eval 1.0

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

      2. cos-atan-rev 1.0

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

      3. cos-atan-rev 1.0

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 0.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.0859375 \cdot \left(x \cdot x\right), -1, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0095:\\ \;\;\;\;\mathsf{fma}\left(\left(0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0095)
   (*
    (fma
     (-
      (*
       (-
        0.07877604166666667
        (*
         (/ (fma (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.5 0.1875) 9.0)
         0.375))
       (* x_m x_m))
      0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m 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 (x_m <= 0.0095) {
		tmp = fma((((0.07877604166666667 - ((fma((sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.5, 0.1875) / 9.0) * 0.375)) * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * 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 (x_m <= 0.0095)
		tmp = Float64(fma(Float64(Float64(Float64(0.07877604166666667 - Float64(Float64(fma(Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.5, 0.1875) / 9.0) * 0.375)) * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * 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[x$95$m, 0.0095], N[(N[(N[(N[(N[(0.07877604166666667 - N[(N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.1875), $MachinePrecision] / 9.0), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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 < 0.00949999999999999976

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{275}{1024} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(-1 \cdot \frac{\left(\frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}\right)}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{3}{16} + \frac{1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) - \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 64.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(0.07877604166666667 - \frac{\mathsf{fma}\left(\sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.5, 0.1875\right)}{9} \cdot 0.375\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.00949999999999999976 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 10: 99.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0028:\\ \;\;\;\;\mathsf{fma}\left(0.0859375 \cdot \left(x\_m \cdot x\_m\right), -1, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0028)
   (* (fma (* 0.0859375 (* x_m x_m)) -1.0 0.125) (* x_m 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 (x_m <= 0.0028) {
		tmp = fma((0.0859375 * (x_m * x_m)), -1.0, 0.125) * (x_m * 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 (x_m <= 0.0028)
		tmp = Float64(fma(Float64(0.0859375 * Float64(x_m * x_m)), -1.0, 0.125) * Float64(x_m * 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[x$95$m, 0.0028], N[(N[(N[(0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $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 < 0.00279999999999999997

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.0859375 \cdot \left(x \cdot x\right), -1, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 0.00279999999999999997 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Add Preprocessing
   3. 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 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 11: 98.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(0.0859375 \cdot \left(x\_m \cdot x\_m\right), -1, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \sqrt{0.125}}{\sqrt{0.5} + 1.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma (* 0.0859375 (* x_m x_m)) -1.0 0.125) (* x_m x_m))
   (/ (- 1.0 (sqrt 0.125)) (+ (sqrt 0.5) 1.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma((0.0859375 * (x_m * x_m)), -1.0, 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - sqrt(0.125)) / (sqrt(0.5) + 1.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(Float64(0.0859375 * Float64(x_m * x_m)), -1.0, 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - sqrt(0.125)) / Float64(sqrt(0.5) + 1.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * -1.0 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Sqrt[0.125], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 67.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 35.9

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

   5. Applied rewrites 35.9%

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

      1. metadata-eval 35.9

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

      2. cos-atan-rev 35.9

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

      3. cos-atan-rev 35.9

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \color{blue}{{x}^{2}} \]

   10. Applied rewrites 63.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.0859375 \cdot \left(x \cdot x\right), -1, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   4. 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 96.8

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

   5. Applied rewrites 96.8%

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

      1. metadata-eval 96.8

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

      2. cos-atan-rev 96.8

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

      3. cos-atan-rev 96.8

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1 - \sqrt{\frac{1}{8}}}{\frac{3}{2} + \sqrt{\frac{1}{2}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - \sqrt{\frac{1}{8}}}{\color{blue}{\frac{3}{2} + \sqrt{\frac{1}{2}}}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1 - \sqrt{\frac{1}{8}}}{\color{blue}{\frac{3}{2}} + \sqrt{\frac{1}{2}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 - \sqrt{\frac{1}{8}}}{\frac{3}{2} + \sqrt{\frac{1}{2}}} \]

      4. +-commutative N/A

         \[\leadsto \frac{1 - \sqrt{\frac{1}{8}}}{\sqrt{\frac{1}{2}} + \color{blue}{\frac{3}{2}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1 - \sqrt{\frac{1}{8}}}{\sqrt{\frac{1}{2}} + \color{blue}{\frac{3}{2}}} \]

      6. lift-sqrt.f64 96.0

         \[\leadsto \frac{1 - \sqrt{0.125}}{\sqrt{0.5} + 1.5} \]

   10. Applied rewrites 96.0%

       \[\leadsto \color{blue}{\frac{1 - \sqrt{0.125}}{\sqrt{0.5} + 1.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 97.5% accurate, 6.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} 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 <= 1.55d0) then
        tmp = 0.125d0 * (x_m * x_m)
    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 <= 1.55) {
		tmp = 0.125 * (x_m * x_m);
	} 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 <= 1.55:
		tmp = 0.125 * (x_m * x_m)
	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 <= 1.55)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	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 <= 1.55)
		tmp = 0.125 * (x_m * x_m);
	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, 1.55], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 67.3%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{8} \cdot {x}^{\color{blue}{2}} \]

      2. pow2 N/A

         \[\leadsto \frac{1}{8} \cdot \left(x \cdot x\right) \]

      3. lift-*.f64 64.1

         \[\leadsto 0.125 \cdot \left(x \cdot x\right) \]

   8. Applied rewrites 64.1%

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

   if 1.55000000000000004 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.6%

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

Alternative 13: 51.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.125 \cdot \left(x\_m \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* 0.125 (* x_m x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return 0.125 * (x_m * x_m);
}

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 = 0.125d0 * (x_m * x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.125 * (x_m * x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.125 * (x_m * x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(0.125 * Float64(x_m * x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.125 * (x_m * x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. sqrt-undiv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 25.8

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

5. Applied rewrites 25.8%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{8} \cdot {x}^{\color{blue}{2}} \]

   2. pow2 N/A

      \[\leadsto \frac{1}{8} \cdot \left(x \cdot x\right) \]

   3. lift-*.f64 50.1

      \[\leadsto 0.125 \cdot \left(x \cdot x\right) \]

8. Applied rewrites 50.1%

   \[\leadsto 0.125 \cdot \color{blue}{\left(x \cdot x\right)} \]
9. Add Preprocessing

Alternative 14: 27.2% 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 74.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
4. 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 24.7

      \[\leadsto 0 \]

5. Applied rewrites 24.7%

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

Reproduce

herbie shell --seed 2025040 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))