Given's Rotation SVD example, simplified

Percentage Accurate: 75.3% → 99.9%

Time: 9.6s

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: 75.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}}\\ t_1 := \mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)\\ t_2 := \mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\\ t_3 := {t\_2}^{2}\\ t_4 := \mathsf{fma}\left(\frac{t\_1}{t\_3}, 0.375, \frac{0.3046875}{t\_2}\right)\\ t_5 := \frac{0.2685546875}{t\_2} - \mathsf{fma}\left(t\_4, \frac{t\_1}{-t\_2}, \frac{\mathsf{fma}\left(t\_0, 0.375, 0.0703125\right)}{t\_3}\right)\\ t_6 := \cos \tan^{-1} x\_m\\ t_7 := \mathsf{fma}\left(0.5, t\_6, 0.5\right)\\ \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \frac{\mathsf{fma}\left(-\left(t\_0 - -0.1875\right), t\_4, 0.245452880859375\right)}{t\_2} + \mathsf{fma}\left(t\_1, \frac{t\_5}{t\_2}, \frac{\frac{-0.134765625 \cdot \sqrt{0.5}}{\sqrt{2}} - 0.15625}{t\_3} \cdot 0.375\right), t\_5\right) \cdot \left(x\_m \cdot x\_m\right) - t\_4, x\_m \cdot x\_m, \frac{0.375}{t\_2}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {t\_7}^{4.5}}{\left({t\_7}^{3} + {t\_7}^{1.5}\right) + 1}}{\mathsf{fma}\left(t\_6 + 1, 0.5, \sqrt{\mathsf{fma}\left(t\_6, 0.5, 0.5\right)}\right) + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ (* 0.171875 (sqrt 0.5)) (sqrt 2.0)))
        (t_1 (fma (/ (sqrt 0.5) (sqrt 2.0)) -0.25 -0.25))
        (t_2 (fma (sqrt 2.0) (sqrt 0.5) 2.0))
        (t_3 (pow t_2 2.0))
        (t_4 (fma (/ t_1 t_3) 0.375 (/ 0.3046875 t_2)))
        (t_5
         (-
          (/ 0.2685546875 t_2)
          (fma t_4 (/ t_1 (- t_2)) (/ (fma t_0 0.375 0.0703125) t_3))))
        (t_6 (cos (atan x_m)))
        (t_7 (fma 0.5 t_6 0.5)))
   (if (<= x_m 0.029)
     (*
      (fma
       (-
        (*
         (fma
          (* (- x_m) x_m)
          (+
           (/ (fma (- (- t_0 -0.1875)) t_4 0.245452880859375) t_2)
           (fma
            t_1
            (/ t_5 t_2)
            (*
             (/ (- (/ (* -0.134765625 (sqrt 0.5)) (sqrt 2.0)) 0.15625) t_3)
             0.375)))
          t_5)
         (* x_m x_m))
        t_4)
       (* x_m x_m)
       (/ 0.375 t_2))
      (* x_m x_m))
     (/
      (/ (- 1.0 (pow t_7 4.5)) (+ (+ (pow t_7 3.0) (pow t_7 1.5)) 1.0))
      (+ (fma (+ t_6 1.0) 0.5 (sqrt (fma t_6 0.5 0.5))) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.171875 * sqrt(0.5)) / sqrt(2.0);
	double t_1 = fma((sqrt(0.5) / sqrt(2.0)), -0.25, -0.25);
	double t_2 = fma(sqrt(2.0), sqrt(0.5), 2.0);
	double t_3 = pow(t_2, 2.0);
	double t_4 = fma((t_1 / t_3), 0.375, (0.3046875 / t_2));
	double t_5 = (0.2685546875 / t_2) - fma(t_4, (t_1 / -t_2), (fma(t_0, 0.375, 0.0703125) / t_3));
	double t_6 = cos(atan(x_m));
	double t_7 = fma(0.5, t_6, 0.5);
	double tmp;
	if (x_m <= 0.029) {
		tmp = fma(((fma((-x_m * x_m), ((fma(-(t_0 - -0.1875), t_4, 0.245452880859375) / t_2) + fma(t_1, (t_5 / t_2), (((((-0.134765625 * sqrt(0.5)) / sqrt(2.0)) - 0.15625) / t_3) * 0.375))), t_5) * (x_m * x_m)) - t_4), (x_m * x_m), (0.375 / t_2)) * (x_m * x_m);
	} else {
		tmp = ((1.0 - pow(t_7, 4.5)) / ((pow(t_7, 3.0) + pow(t_7, 1.5)) + 1.0)) / (fma((t_6 + 1.0), 0.5, sqrt(fma(t_6, 0.5, 0.5))) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.171875 * sqrt(0.5)) / sqrt(2.0))
	t_1 = fma(Float64(sqrt(0.5) / sqrt(2.0)), -0.25, -0.25)
	t_2 = fma(sqrt(2.0), sqrt(0.5), 2.0)
	t_3 = t_2 ^ 2.0
	t_4 = fma(Float64(t_1 / t_3), 0.375, Float64(0.3046875 / t_2))
	t_5 = Float64(Float64(0.2685546875 / t_2) - fma(t_4, Float64(t_1 / Float64(-t_2)), Float64(fma(t_0, 0.375, 0.0703125) / t_3)))
	t_6 = cos(atan(x_m))
	t_7 = fma(0.5, t_6, 0.5)
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(-x_m) * x_m), Float64(Float64(fma(Float64(-Float64(t_0 - -0.1875)), t_4, 0.245452880859375) / t_2) + fma(t_1, Float64(t_5 / t_2), Float64(Float64(Float64(Float64(Float64(-0.134765625 * sqrt(0.5)) / sqrt(2.0)) - 0.15625) / t_3) * 0.375))), t_5) * Float64(x_m * x_m)) - t_4), Float64(x_m * x_m), Float64(0.375 / t_2)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_7 ^ 4.5)) / Float64(Float64((t_7 ^ 3.0) + (t_7 ^ 1.5)) + 1.0)) / Float64(fma(Float64(t_6 + 1.0), 0.5, sqrt(fma(t_6, 0.5, 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[(0.171875 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.25 + -0.25), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 / t$95$3), $MachinePrecision] * 0.375 + N[(0.3046875 / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.2685546875 / t$95$2), $MachinePrecision] - N[(t$95$4 * N[(t$95$1 / (-t$95$2)), $MachinePrecision] + N[(N[(t$95$0 * 0.375 + 0.0703125), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(0.5 * t$95$6 + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.029], N[(N[(N[(N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[((-N[(t$95$0 - -0.1875), $MachinePrecision]) * t$95$4 + 0.245452880859375), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(t$95$1 * N[(t$95$5 / t$95$2), $MachinePrecision] + N[(N[(N[(N[(N[(-0.134765625 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] - 0.15625), $MachinePrecision] / t$95$3), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.375 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$7, 4.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$7, 3.0], $MachinePrecision] + N[Power[t$95$7, 1.5], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$6 + 1.0), $MachinePrecision] * 0.5 + N[Sqrt[N[(t$95$6 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 69.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.7%

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 67.7%

       \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]

   11. 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)} \]

   13. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \frac{\mathsf{fma}\left(-1 \cdot \left(\frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}} - -0.1875\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), 0.245452880859375\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right), \frac{\frac{0.2685546875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), -\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}, \frac{\mathsf{fma}\left(\frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}}, 0.375, 0.0703125\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}, \frac{\frac{-0.134765625 \cdot \sqrt{0.5}}{\sqrt{2}} - 0.15625}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}} \cdot 0.375\right), \frac{0.2685546875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), -\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}, \frac{\mathsf{fma}\left(\frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}}, 0.375, 0.0703125\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}\right)\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), x \cdot x, \frac{0.375}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right) \cdot \left(x \cdot x\right)} \]

   if 0.0290000000000000015 < x

   1. Initial program 98.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.029:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(-x\right) \cdot x, \frac{\mathsf{fma}\left(-\left(\frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}} - -0.1875\right), \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), 0.245452880859375\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right), \frac{\frac{0.2685546875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{-\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}, \frac{\mathsf{fma}\left(\frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}}, 0.375, 0.0703125\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}, \frac{\frac{-0.134765625 \cdot \sqrt{0.5}}{\sqrt{2}} - 0.15625}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}} \cdot 0.375\right), \frac{0.2685546875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)} - \mathsf{fma}\left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{-\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}, \frac{\mathsf{fma}\left(\frac{0.171875 \cdot \sqrt{0.5}}{\sqrt{2}}, 0.375, 0.0703125\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}\right)\right) \cdot \left(x \cdot x\right) - \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), x \cdot x, \frac{0.375}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(\mathsf{fma}\left(0.5, \cos \tan^{-1} x, 0.5\right)\right)}^{4.5}}{\left({\left(\mathsf{fma}\left(0.5, \cos \tan^{-1} x, 0.5\right)\right)}^{3} + {\left(\mathsf{fma}\left(0.5, \cos \tan^{-1} x, 0.5\right)\right)}^{1.5}\right) + 1}}{\mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := \sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right)}\\ t_2 := t\_1 - -1\\ t_3 := \mathsf{fma}\left(0.5, t\_0, 0.5\right)\\ \mathbf{if}\;x\_m \leq 0.0245:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right)}{t\_2} \cdot \left(x\_m \cdot x\_m\right) - \frac{0.1875}{t\_2}, x\_m \cdot x\_m, \frac{0.25}{t\_2}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {t\_3}^{4.5}}{\left({t\_3}^{3} + {t\_3}^{1.5}\right) + 1}}{\mathsf{fma}\left(t\_0 + 1, 0.5, t\_1\right) + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m)))
        (t_1 (sqrt (fma t_0 0.5 0.5)))
        (t_2 (- t_1 -1.0))
        (t_3 (fma 0.5 t_0 0.5)))
   (if (<= x_m 0.0245)
     (*
      (fma
       (-
        (* (/ (fma -0.13671875 (* x_m x_m) 0.15625) t_2) (* x_m x_m))
        (/ 0.1875 t_2))
       (* x_m x_m)
       (/ 0.25 t_2))
      (* x_m x_m))
     (/
      (/ (- 1.0 (pow t_3 4.5)) (+ (+ (pow t_3 3.0) (pow t_3 1.5)) 1.0))
      (+ (fma (+ t_0 1.0) 0.5 t_1) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = sqrt(fma(t_0, 0.5, 0.5));
	double t_2 = t_1 - -1.0;
	double t_3 = fma(0.5, t_0, 0.5);
	double tmp;
	if (x_m <= 0.0245) {
		tmp = fma((((fma(-0.13671875, (x_m * x_m), 0.15625) / t_2) * (x_m * x_m)) - (0.1875 / t_2)), (x_m * x_m), (0.25 / t_2)) * (x_m * x_m);
	} else {
		tmp = ((1.0 - pow(t_3, 4.5)) / ((pow(t_3, 3.0) + pow(t_3, 1.5)) + 1.0)) / (fma((t_0 + 1.0), 0.5, t_1) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = sqrt(fma(t_0, 0.5, 0.5))
	t_2 = Float64(t_1 - -1.0)
	t_3 = fma(0.5, t_0, 0.5)
	tmp = 0.0
	if (x_m <= 0.0245)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.13671875, Float64(x_m * x_m), 0.15625) / t_2) * Float64(x_m * x_m)) - Float64(0.1875 / t_2)), Float64(x_m * x_m), Float64(0.25 / t_2)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(Float64(1.0 - (t_3 ^ 4.5)) / Float64(Float64((t_3 ^ 3.0) + (t_3 ^ 1.5)) + 1.0)) / Float64(fma(Float64(t_0 + 1.0), 0.5, t_1) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$0 + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0245], N[(N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(0.1875 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.25 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[t$95$3, 4.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] + N[Power[t$95$3, 1.5], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.024500000000000001

   1. Initial program 69.1%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.6

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-35}{256} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}} + \frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right) - \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right)} \]

   9. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.13671875, x \cdot x, 0.15625\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1} \cdot \left(x \cdot x\right) - \frac{0.1875}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1}, x \cdot x, \frac{0.25}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1}\right) \cdot \left(x \cdot x\right)} \]

   if 0.024500000000000001 < x

   1. Initial program 98.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := \mathsf{fma}\left(0.5, t\_0, 0.5\right)\\ t_2 := \sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right)} - -1\\ \mathbf{if}\;x\_m \leq 0.0245:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right)}{t\_2} \cdot \left(x\_m \cdot x\_m\right) - \frac{0.1875}{t\_2}, x\_m \cdot x\_m, \frac{0.25}{t\_2}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_1}^{4.5}}{\left(\left({t\_1}^{3} + {t\_1}^{1.5}\right) + 1\right) \cdot \left(\mathsf{fma}\left(0.5, t\_0, 1.5\right) + \sqrt{t\_1}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m)))
        (t_1 (fma 0.5 t_0 0.5))
        (t_2 (- (sqrt (fma t_0 0.5 0.5)) -1.0)))
   (if (<= x_m 0.0245)
     (*
      (fma
       (-
        (* (/ (fma -0.13671875 (* x_m x_m) 0.15625) t_2) (* x_m x_m))
        (/ 0.1875 t_2))
       (* x_m x_m)
       (/ 0.25 t_2))
      (* x_m x_m))
     (/
      (- 1.0 (pow t_1 4.5))
      (*
       (+ (+ (pow t_1 3.0) (pow t_1 1.5)) 1.0)
       (+ (fma 0.5 t_0 1.5) (sqrt t_1)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = fma(0.5, t_0, 0.5);
	double t_2 = sqrt(fma(t_0, 0.5, 0.5)) - -1.0;
	double tmp;
	if (x_m <= 0.0245) {
		tmp = fma((((fma(-0.13671875, (x_m * x_m), 0.15625) / t_2) * (x_m * x_m)) - (0.1875 / t_2)), (x_m * x_m), (0.25 / t_2)) * (x_m * x_m);
	} else {
		tmp = (1.0 - pow(t_1, 4.5)) / (((pow(t_1, 3.0) + pow(t_1, 1.5)) + 1.0) * (fma(0.5, t_0, 1.5) + sqrt(t_1)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = fma(0.5, t_0, 0.5)
	t_2 = Float64(sqrt(fma(t_0, 0.5, 0.5)) - -1.0)
	tmp = 0.0
	if (x_m <= 0.0245)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.13671875, Float64(x_m * x_m), 0.15625) / t_2) * Float64(x_m * x_m)) - Float64(0.1875 / t_2)), Float64(x_m * x_m), Float64(0.25 / t_2)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 4.5)) / Float64(Float64(Float64((t_1 ^ 3.0) + (t_1 ^ 1.5)) + 1.0) * Float64(fma(0.5, t_0, 1.5) + sqrt(t_1))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0245], N[(N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(0.1875 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.25 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 4.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(0.5 * t$95$0 + 1.5), $MachinePrecision] + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.024500000000000001

   1. Initial program 69.1%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.6

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-35}{256} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}} + \frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right) - \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right)} \]

   9. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.13671875, x \cdot x, 0.15625\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1} \cdot \left(x \cdot x\right) - \frac{0.1875}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1}, x \cdot x, \frac{0.25}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1}\right) \cdot \left(x \cdot x\right)} \]

   if 0.024500000000000001 < x

   1. Initial program 98.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := \sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right)}\\ t_2 := t\_1 - -1\\ \mathbf{if}\;x\_m \leq 0.024:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right)}{t\_2} \cdot \left(x\_m \cdot x\_m\right) - \frac{0.1875}{t\_2}, x\_m \cdot x\_m, \frac{0.25}{t\_2}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}, 0.5, 0.5\right)\right)}^{1.5}}{\mathsf{fma}\left(t\_0 + 1, 0.5, t\_1\right) + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m)))
        (t_1 (sqrt (fma t_0 0.5 0.5)))
        (t_2 (- t_1 -1.0)))
   (if (<= x_m 0.024)
     (*
      (fma
       (-
        (* (/ (fma -0.13671875 (* x_m x_m) 0.15625) t_2) (* x_m x_m))
        (/ 0.1875 t_2))
       (* x_m x_m)
       (/ 0.25 t_2))
      (* x_m x_m))
     (/
      (- 1.0 (pow (fma (sqrt (/ 1.0 (fma x_m x_m 1.0))) 0.5 0.5) 1.5))
      (+ (fma (+ t_0 1.0) 0.5 t_1) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = sqrt(fma(t_0, 0.5, 0.5));
	double t_2 = t_1 - -1.0;
	double tmp;
	if (x_m <= 0.024) {
		tmp = fma((((fma(-0.13671875, (x_m * x_m), 0.15625) / t_2) * (x_m * x_m)) - (0.1875 / t_2)), (x_m * x_m), (0.25 / t_2)) * (x_m * x_m);
	} else {
		tmp = (1.0 - pow(fma(sqrt((1.0 / fma(x_m, x_m, 1.0))), 0.5, 0.5), 1.5)) / (fma((t_0 + 1.0), 0.5, t_1) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = sqrt(fma(t_0, 0.5, 0.5))
	t_2 = Float64(t_1 - -1.0)
	tmp = 0.0
	if (x_m <= 0.024)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.13671875, Float64(x_m * x_m), 0.15625) / t_2) * Float64(x_m * x_m)) - Float64(0.1875 / t_2)), Float64(x_m * x_m), Float64(0.25 / t_2)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - (fma(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))), 0.5, 0.5) ^ 1.5)) / Float64(fma(Float64(t_0 + 1.0), 0.5, t_1) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - -1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.024], N[(N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(0.1875 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.25 / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.024

   1. Initial program 69.1%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.6

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-35}{256} \cdot \frac{{x}^{2}}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}} + \frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right) - \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \tan^{-1} x}}\right)} \]

   9. Applied rewrites 68.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.13671875, x \cdot x, 0.15625\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1} \cdot \left(x \cdot x\right) - \frac{0.1875}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1}, x \cdot x, \frac{0.25}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} - -1}\right) \cdot \left(x \cdot x\right)} \]

   if 0.024 < x

   1. Initial program 98.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := \sqrt{\mathsf{fma}\left(t\_0, 0.5, 0.5\right)}\\ \mathbf{if}\;x\_m \leq 0.024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right), x\_m \cdot x\_m, -0.1875\right), x\_m \cdot x\_m, 0.25\right) \cdot \left(x\_m \cdot x\_m\right)}{t\_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}, 0.5, 0.5\right)\right)}^{1.5}}{\mathsf{fma}\left(t\_0 + 1, 0.5, t\_1\right) + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m))) (t_1 (sqrt (fma t_0 0.5 0.5))))
   (if (<= x_m 0.024)
     (/
      (*
       (fma
        (fma (fma -0.13671875 (* x_m x_m) 0.15625) (* x_m x_m) -0.1875)
        (* x_m x_m)
        0.25)
       (* x_m x_m))
      (+ t_1 1.0))
     (/
      (- 1.0 (pow (fma (sqrt (/ 1.0 (fma x_m x_m 1.0))) 0.5 0.5) 1.5))
      (+ (fma (+ t_0 1.0) 0.5 t_1) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = sqrt(fma(t_0, 0.5, 0.5));
	double tmp;
	if (x_m <= 0.024) {
		tmp = (fma(fma(fma(-0.13671875, (x_m * x_m), 0.15625), (x_m * x_m), -0.1875), (x_m * x_m), 0.25) * (x_m * x_m)) / (t_1 + 1.0);
	} else {
		tmp = (1.0 - pow(fma(sqrt((1.0 / fma(x_m, x_m, 1.0))), 0.5, 0.5), 1.5)) / (fma((t_0 + 1.0), 0.5, t_1) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = sqrt(fma(t_0, 0.5, 0.5))
	tmp = 0.0
	if (x_m <= 0.024)
		tmp = Float64(Float64(fma(fma(fma(-0.13671875, Float64(x_m * x_m), 0.15625), Float64(x_m * x_m), -0.1875), Float64(x_m * x_m), 0.25) * Float64(x_m * x_m)) / Float64(t_1 + 1.0));
	else
		tmp = Float64(Float64(1.0 - (fma(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))), 0.5, 0.5) ^ 1.5)) / Float64(fma(Float64(t_0 + 1.0), 0.5, t_1) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$95$m, 0.024], N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.1875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.024

   1. Initial program 69.1%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.6

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{32} + \frac{-35}{256} \cdot {x}^{2}\right) - \frac{3}{16}\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, \frac{1}{2}, \frac{1}{2}\right)} + 1} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x \cdot x, 0.15625\right), x \cdot x, -0.1875\right), x \cdot x, 0.25\right) \cdot \left(x \cdot x\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} + 1} \]

   if 0.024 < x

   1. Initial program 98.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 6: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(\cos \tan^{-1} x\_m, 0.5, 0.5\right)} + 1\\ \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right), x\_m \cdot x\_m, -0.1875\right), x\_m \cdot x\_m, 0.25\right) \cdot \left(x\_m \cdot x\_m\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left({\left(\mathsf{fma}\left(x\_m, x\_m, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (sqrt (fma (cos (atan x_m)) 0.5 0.5)) 1.0)))
   (if (<= x_m 0.029)
     (/
      (*
       (fma
        (fma (fma -0.13671875 (* x_m x_m) 0.15625) (* x_m x_m) -0.1875)
        (* x_m x_m)
        0.25)
       (* x_m x_m))
      t_0)
     (/ (- 1.0 (fma (pow (fma x_m x_m 1.0) -0.5) 0.5 0.5)) t_0))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0;
	double tmp;
	if (x_m <= 0.029) {
		tmp = (fma(fma(fma(-0.13671875, (x_m * x_m), 0.15625), (x_m * x_m), -0.1875), (x_m * x_m), 0.25) * (x_m * x_m)) / t_0;
	} else {
		tmp = (1.0 - fma(pow(fma(x_m, x_m, 1.0), -0.5), 0.5, 0.5)) / t_0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0)
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(Float64(fma(fma(fma(-0.13671875, Float64(x_m * x_m), 0.15625), Float64(x_m * x_m), -0.1875), Float64(x_m * x_m), 0.25) * Float64(x_m * x_m)) / t_0);
	else
		tmp = Float64(Float64(1.0 - fma((fma(x_m, x_m, 1.0) ^ -0.5), 0.5, 0.5)) / t_0);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.029], N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.1875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 - N[(N[Power[N[(x$95$m * x$95$m + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 69.1%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.6

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{32} + \frac{-35}{256} \cdot {x}^{2}\right) - \frac{3}{16}\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, \frac{1}{2}, \frac{1}{2}\right)} + 1} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x \cdot x, 0.15625\right), x \cdot x, -0.1875\right), x \cdot x, 0.25\right) \cdot \left(x \cdot x\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} + 1} \]

   if 0.0290000000000000015 < x

   1. Initial program 98.2%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 7: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5\\ \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right), x\_m \cdot x\_m, -0.1875\right), x\_m \cdot x\_m, 0.25\right) \cdot \left(x\_m \cdot x\_m\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x\_m, 0.5, 0.5\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))
   (if (<= x_m 1.0)
     (/
      (*
       (fma
        (fma (fma -0.13671875 (* x_m x_m) 0.15625) (* x_m x_m) -0.1875)
        (* x_m x_m)
        0.25)
       (* x_m x_m))
      (+ (sqrt (fma (cos (atan x_m)) 0.5 0.5)) 1.0))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5;
	double tmp;
	if (x_m <= 1.0) {
		tmp = (fma(fma(fma(-0.13671875, (x_m * x_m), 0.15625), (x_m * x_m), -0.1875), (x_m * x_m), 0.25) * (x_m * x_m)) / (sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0);
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(fma(fma(fma(-0.13671875, Float64(x_m * x_m), 0.15625), Float64(x_m * x_m), -0.1875), Float64(x_m * x_m), 0.25) * Float64(x_m * x_m)) / Float64(sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.0], N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.1875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 69.2%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{32} + \frac{-35}{256} \cdot {x}^{2}\right) - \frac{3}{16}\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, \frac{1}{2}, \frac{1}{2}\right)} + 1} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x \cdot x, 0.15625\right), x \cdot x, -0.1875\right), x \cdot x, 0.25\right) \cdot \left(x \cdot x\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} + 1} \]

   if 1 < 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

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

   5. Applied rewrites 97.7%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5 - \frac{0.25}{x \cdot x}}{x} - -0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 99.2%

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

Alternative 8: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5\\ \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.15625, x\_m \cdot x\_m, -0.1875\right), x\_m \cdot x\_m, 0.25\right) \cdot \left(x\_m \cdot x\_m\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x\_m, 0.5, 0.5\right)} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))
   (if (<= x_m 1.05)
     (/
      (* (fma (fma 0.15625 (* x_m x_m) -0.1875) (* x_m x_m) 0.25) (* x_m x_m))
      (+ (sqrt (fma (cos (atan x_m)) 0.5 0.5)) 1.0))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5;
	double tmp;
	if (x_m <= 1.05) {
		tmp = (fma(fma(0.15625, (x_m * x_m), -0.1875), (x_m * x_m), 0.25) * (x_m * x_m)) / (sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0);
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(Float64(fma(fma(0.15625, Float64(x_m * x_m), -0.1875), Float64(x_m * x_m), 0.25) * Float64(x_m * x_m)) / Float64(sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05], N[(N[(N[(N[(0.15625 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.1875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 69.2%

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

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 69.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 69.7

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

   6. Applied rewrites 69.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{5}{32} \cdot {x}^{2} - \frac{3}{16}\right)\right)}}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, \frac{1}{2}, \frac{1}{2}\right)} + 1} \]
   8. Step-by-step derivation

   9. Applied rewrites 68.5%

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

   if 1.05000000000000004 < 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

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

   5. Applied rewrites 97.7%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5 - \frac{0.25}{x \cdot x}}{x} - -0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 99.2%

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

Alternative 9: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5\\ t_1 := \mathsf{fma}\left(-0.25, x\_m \cdot x\_m, 1\right)\\ \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2685546875, x\_m \cdot x\_m, -0.3046875\right), x\_m \cdot x\_m, 0.375\right) \cdot \left(x\_m \cdot x\_m\right)}{\left(1 + t\_1\right) + \sqrt{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5))
        (t_1 (fma -0.25 (* x_m x_m) 1.0)))
   (if (<= x_m 1.0)
     (/
      (*
       (fma (fma 0.2685546875 (* x_m x_m) -0.3046875) (* x_m x_m) 0.375)
       (* x_m x_m))
      (+ (+ 1.0 t_1) (sqrt t_1)))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5;
	double t_1 = fma(-0.25, (x_m * x_m), 1.0);
	double tmp;
	if (x_m <= 1.0) {
		tmp = (fma(fma(0.2685546875, (x_m * x_m), -0.3046875), (x_m * x_m), 0.375) * (x_m * x_m)) / ((1.0 + t_1) + sqrt(t_1));
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)
	t_1 = fma(-0.25, Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(Float64(fma(fma(0.2685546875, Float64(x_m * x_m), -0.3046875), Float64(x_m * x_m), 0.375) * Float64(x_m * x_m)) / Float64(Float64(1.0 + t_1) + sqrt(t_1)));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.0], N[(N[(N[(N[(0.2685546875 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.3046875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$1), $MachinePrecision] + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)\right)}}{\left(1 + \mathsf{fma}\left(\frac{-1}{4}, x \cdot x, 1\right)\right) + \sqrt{\mathsf{fma}\left(\frac{-1}{4}, x \cdot x, 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2685546875, x \cdot x, -0.3046875\right), x \cdot x, 0.375\right) \cdot \left(x \cdot x\right)}}{\left(1 + \mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right) + \sqrt{\mathsf{fma}\left(-0.25, x \cdot x, 1\right)}} \]

   if 1 < 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

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

   5. Applied rewrites 97.7%

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{0.5 - \frac{0.25}{x \cdot x}}{x} - -0.5}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 99.2%

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

Alternative 10: 98.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.25, x\_m \cdot x\_m, 1\right)\\ \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2685546875, x\_m \cdot x\_m, -0.3046875\right), x\_m \cdot x\_m, 0.375\right) \cdot \left(x\_m \cdot x\_m\right)}{\left(1 + t\_0\right) + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma -0.25 (* x_m x_m) 1.0)))
   (if (<= x_m 1.2)
     (/
      (*
       (fma (fma 0.2685546875 (* x_m x_m) -0.3046875) (* x_m x_m) 0.375)
       (* x_m x_m))
      (+ (+ 1.0 t_0) (sqrt t_0)))
     (/ 0.5 (- (sqrt 0.5) -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(-0.25, (x_m * x_m), 1.0);
	double tmp;
	if (x_m <= 1.2) {
		tmp = (fma(fma(0.2685546875, (x_m * x_m), -0.3046875), (x_m * x_m), 0.375) * (x_m * x_m)) / ((1.0 + t_0) + sqrt(t_0));
	} else {
		tmp = 0.5 / (sqrt(0.5) - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(-0.25, Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(Float64(fma(fma(0.2685546875, Float64(x_m * x_m), -0.3046875), Float64(x_m * x_m), 0.375) * Float64(x_m * x_m)) / Float64(Float64(1.0 + t_0) + sqrt(t_0)));
	else
		tmp = Float64(0.5 / Float64(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[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.2], N[(N[(N[(N[(0.2685546875 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.3046875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$0), $MachinePrecision] + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(\frac{3}{8} + {x}^{2} \cdot \left(\frac{275}{1024} \cdot {x}^{2} - \frac{39}{128}\right)\right)}}{\left(1 + \mathsf{fma}\left(\frac{-1}{4}, x \cdot x, 1\right)\right) + \sqrt{\mathsf{fma}\left(\frac{-1}{4}, x \cdot x, 1\right)}} \]
   9. Step-by-step derivation

   10. Applied rewrites 67.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2685546875, x \cdot x, -0.3046875\right), x \cdot x, 0.375\right) \cdot \left(x \cdot x\right)}}{\left(1 + \mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right) + \sqrt{\mathsf{fma}\left(-0.25, x \cdot x, 1\right)}} \]

   if 1.19999999999999996 < 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 0

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

   5. Applied rewrites 0.0%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.8%

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

Alternative 11: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.25, x\_m \cdot x\_m, 1\right)\\ \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.3046875, x\_m \cdot x\_m, 0.375\right) \cdot \left(x\_m \cdot x\_m\right)}{\left(1 + t\_0\right) + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma -0.25 (* x_m x_m) 1.0)))
   (if (<= x_m 1.05)
     (/
      (* (fma -0.3046875 (* x_m x_m) 0.375) (* x_m x_m))
      (+ (+ 1.0 t_0) (sqrt t_0)))
     (/ 0.5 (- (sqrt 0.5) -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(-0.25, (x_m * x_m), 1.0);
	double tmp;
	if (x_m <= 1.05) {
		tmp = (fma(-0.3046875, (x_m * x_m), 0.375) * (x_m * x_m)) / ((1.0 + t_0) + sqrt(t_0));
	} else {
		tmp = 0.5 / (sqrt(0.5) - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(-0.25, Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(Float64(fma(-0.3046875, Float64(x_m * x_m), 0.375) * Float64(x_m * x_m)) / Float64(Float64(1.0 + t_0) + sqrt(t_0)));
	else
		tmp = Float64(0.5 / Float64(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[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05], N[(N[(N[(-0.3046875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$0), $MachinePrecision] + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 66.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.3046875, x \cdot x, 0.375\right) \cdot \left(x \cdot x\right)}}{\left(1 + \mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right) + \sqrt{\mathsf{fma}\left(-0.25, x \cdot x, 1\right)}} \]

   if 1.05000000000000004 < 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 0

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

   5. Applied rewrites 0.0%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.8%

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

Alternative 12: 98.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 0.5d0 / (sqrt(0.5d0) - (-1.0d0))
    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 = (x_m * x_m) * 0.125;
	} else {
		tmp = 0.5 / (Math.sqrt(0.5) - -1.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 0.5 / (math.sqrt(0.5) - -1.0)
	return tmp

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 67.5%

       \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 67.6%

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

   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 0

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

   5. Applied rewrites 0.0%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.8%

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

Alternative 13: 97.8% accurate, 6.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} 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 = (x_m * x_m) * 0.125d0
    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 = (x_m * x_m) * 0.125;
	} 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 = (x_m * x_m) * 0.125
	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(Float64(x_m * x_m) * 0.125);
	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 = (x_m * x_m) * 0.125;
	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[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $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 69.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 67.5%

       \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 67.6%

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

   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 97.2%

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

Alternative 14: 52.3% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

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 = (x_m * x_m) * 0.125d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 27.9%

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

7. Applied rewrites 27.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 52.4%

    \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 52.4%

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

Reproduce

herbie shell --seed 2025024 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))