Given's Rotation SVD example, simplified

Percentage Accurate: 76.0% → 99.9%

Time: 6.3s

Alternatives: 12

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: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \frac{1}{t\_0}\\ t_2 := \sqrt{2} \cdot t\_0\\ t_3 := \sqrt{2} \cdot {t\_0}^{2}\\ t_4 := \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{t\_3}, 0.1875 \cdot t\_1\right)\\ t_5 := 0.15625 \cdot t\_1\\ t_6 := {\left(\sqrt{2}\right)}^{2}\\ t_7 := 0.375 - 0.0625 \cdot \frac{1}{t\_6}\\ t_8 := \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot t\_7}{t\_3}, 0.25 \cdot \frac{\sqrt{0.5} \cdot t\_4}{t\_2}\right)\\ t_9 := \cos \tan^{-1} x\_m + 1\\ t_10 := t\_9 \cdot 0.5\\ t_11 := 1 + \mathsf{fma}\left(t\_9, 0.5, \sqrt{t\_10}\right)\\ \mathbf{if}\;x\_m \leq 0.0295:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left({x\_m}^{2}, {x\_m}^{2} \cdot \left(\mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.5, \frac{\sqrt{0.5} \cdot \left(t\_4 \cdot t\_7\right)}{t\_2}, \mathsf{fma}\left(-0.25, \frac{\sqrt{0.5} \cdot \left(t\_5 - t\_8\right)}{t\_2}, \mathsf{fma}\left(-0.125, \frac{\sqrt{0.5} \cdot \left(0.3125 + -0.25 \cdot \frac{t\_7}{t\_6}\right)}{t\_3}, 0.13671875 \cdot t\_1\right)\right)\right), t\_5\right) - t\_8\right) - t\_4, 0.25 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_11} - \frac{{t\_10}^{1.5}}{t\_11}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (/ 1.0 t_0))
        (t_2 (* (sqrt 2.0) t_0))
        (t_3 (* (sqrt 2.0) (pow t_0 2.0)))
        (t_4 (fma -0.0625 (/ (sqrt 0.5) t_3) (* 0.1875 t_1)))
        (t_5 (* 0.15625 t_1))
        (t_6 (pow (sqrt 2.0) 2.0))
        (t_7 (- 0.375 (* 0.0625 (/ 1.0 t_6))))
        (t_8
         (fma
          0.125
          (/ (* (sqrt 0.5) t_7) t_3)
          (* 0.25 (/ (* (sqrt 0.5) t_4) t_2))))
        (t_9 (+ (cos (atan x_m)) 1.0))
        (t_10 (* t_9 0.5))
        (t_11 (+ 1.0 (fma t_9 0.5 (sqrt t_10)))))
   (if (<= x_m 0.0295)
     (*
      (pow x_m 2.0)
      (fma
       (pow x_m 2.0)
       (-
        (*
         (pow x_m 2.0)
         (-
          (fma
           -1.0
           (*
            (pow x_m 2.0)
            (fma
             -0.5
             (/ (* (sqrt 0.5) (* t_4 t_7)) t_2)
             (fma
              -0.25
              (/ (* (sqrt 0.5) (- t_5 t_8)) t_2)
              (fma
               -0.125
               (/ (* (sqrt 0.5) (+ 0.3125 (* -0.25 (/ t_7 t_6)))) t_3)
               (* 0.13671875 t_1)))))
           t_5)
          t_8))
        t_4)
       (* 0.25 t_1)))
     (- (/ 1.0 t_11) (/ (pow t_10 1.5) t_11)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = 1.0 / t_0;
	double t_2 = sqrt(2.0) * t_0;
	double t_3 = sqrt(2.0) * pow(t_0, 2.0);
	double t_4 = fma(-0.0625, (sqrt(0.5) / t_3), (0.1875 * t_1));
	double t_5 = 0.15625 * t_1;
	double t_6 = pow(sqrt(2.0), 2.0);
	double t_7 = 0.375 - (0.0625 * (1.0 / t_6));
	double t_8 = fma(0.125, ((sqrt(0.5) * t_7) / t_3), (0.25 * ((sqrt(0.5) * t_4) / t_2)));
	double t_9 = cos(atan(x_m)) + 1.0;
	double t_10 = t_9 * 0.5;
	double t_11 = 1.0 + fma(t_9, 0.5, sqrt(t_10));
	double tmp;
	if (x_m <= 0.0295) {
		tmp = pow(x_m, 2.0) * fma(pow(x_m, 2.0), ((pow(x_m, 2.0) * (fma(-1.0, (pow(x_m, 2.0) * fma(-0.5, ((sqrt(0.5) * (t_4 * t_7)) / t_2), fma(-0.25, ((sqrt(0.5) * (t_5 - t_8)) / t_2), fma(-0.125, ((sqrt(0.5) * (0.3125 + (-0.25 * (t_7 / t_6)))) / t_3), (0.13671875 * t_1))))), t_5) - t_8)) - t_4), (0.25 * t_1));
	} else {
		tmp = (1.0 / t_11) - (pow(t_10, 1.5) / t_11);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(sqrt(2.0) * t_0)
	t_3 = Float64(sqrt(2.0) * (t_0 ^ 2.0))
	t_4 = fma(-0.0625, Float64(sqrt(0.5) / t_3), Float64(0.1875 * t_1))
	t_5 = Float64(0.15625 * t_1)
	t_6 = sqrt(2.0) ^ 2.0
	t_7 = Float64(0.375 - Float64(0.0625 * Float64(1.0 / t_6)))
	t_8 = fma(0.125, Float64(Float64(sqrt(0.5) * t_7) / t_3), Float64(0.25 * Float64(Float64(sqrt(0.5) * t_4) / t_2)))
	t_9 = Float64(cos(atan(x_m)) + 1.0)
	t_10 = Float64(t_9 * 0.5)
	t_11 = Float64(1.0 + fma(t_9, 0.5, sqrt(t_10)))
	tmp = 0.0
	if (x_m <= 0.0295)
		tmp = Float64((x_m ^ 2.0) * fma((x_m ^ 2.0), Float64(Float64((x_m ^ 2.0) * Float64(fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.5, Float64(Float64(sqrt(0.5) * Float64(t_4 * t_7)) / t_2), fma(-0.25, Float64(Float64(sqrt(0.5) * Float64(t_5 - t_8)) / t_2), fma(-0.125, Float64(Float64(sqrt(0.5) * Float64(0.3125 + Float64(-0.25 * Float64(t_7 / t_6)))) / t_3), Float64(0.13671875 * t_1))))), t_5) - t_8)) - t_4), Float64(0.25 * t_1)));
	else
		tmp = Float64(Float64(1.0 / t_11) - Float64((t_10 ^ 1.5) / t_11));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.1875 * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.15625 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$7 = N[(0.375 - N[(0.0625 * N[(1.0 / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(0.125 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$7), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.25 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$10 = N[(t$95$9 * 0.5), $MachinePrecision]}, Block[{t$95$11 = N[(1.0 + N[(t$95$9 * 0.5 + N[Sqrt[t$95$10], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0295], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.5 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(t$95$4 * t$95$7), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(-0.25 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(t$95$5 - t$95$8), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(-0.125 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.3125 + N[(-0.25 * N[(t$95$7 / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.13671875 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$5), $MachinePrecision] - t$95$8), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$11), $MachinePrecision] - N[(N[Power[t$95$10, 1.5], $MachinePrecision] / t$95$11), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029499999999999998

   1. Initial program 64.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 27.2

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

   5. Applied rewrites 27.2%

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

      1. metadata-eval 27.2

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

      2. cos-atan-rev 27.2

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

      3. cos-atan-rev 27.2

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)} + \left(\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(\frac{1}{8} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{1}{4} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)} + \left(\frac{-1}{8} \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} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{35}{256} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right)\right)\right) + \frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) - \left(\frac{1}{8} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{1}{4} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right) - \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 72.3%

       \[\leadsto \color{blue}{{x}^{2} \cdot \mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\mathsf{fma}\left(-1, {x}^{2} \cdot \mathsf{fma}\left(-0.5, \frac{\sqrt{0.5} \cdot \left(\mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right) \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}, \mathsf{fma}\left(-0.25, \frac{\sqrt{0.5} \cdot \left(0.15625 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}} - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.25 \cdot \frac{\sqrt{0.5} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}\right)\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}, \mathsf{fma}\left(-0.125, \frac{\sqrt{0.5} \cdot \left(0.3125 + -0.25 \cdot \frac{0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.13671875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)\right)\right), 0.15625 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right) - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.25 \cdot \frac{\sqrt{0.5} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}\right)\right) - \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)} \]

   if 0.029499999999999998 < 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

   4. Applied rewrites 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0295:\\ \;\;\;\;{x}^{2} \cdot \mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\mathsf{fma}\left(-1, {x}^{2} \cdot \mathsf{fma}\left(-0.5, \frac{\sqrt{0.5} \cdot \left(\mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right) \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}, \mathsf{fma}\left(-0.25, \frac{\sqrt{0.5} \cdot \left(0.15625 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}} - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.25 \cdot \frac{\sqrt{0.5} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}\right)\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}, \mathsf{fma}\left(-0.125, \frac{\sqrt{0.5} \cdot \left(0.3125 + -0.25 \cdot \frac{0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.13671875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)\right)\right), 0.15625 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right) - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.25 \cdot \frac{\sqrt{0.5} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}\right)\right) - \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {\left(1 + \sqrt{0.5} \cdot \sqrt{2}\right)}^{2}}, 0.1875 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right), 0.25 \cdot \frac{1}{1 + \sqrt{0.5} \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)} - \frac{{\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{1.5}}{1 + \mathsf{fma}\left(\cos \tan^{-1} x + 1, 0.5, \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \frac{1}{t\_0}\\ t_2 := \sqrt{2} \cdot {t\_0}^{2}\\ t_3 := \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{t\_2}, 0.1875 \cdot t\_1\right)\\ \mathbf{if}\;x\_m \leq 0.008:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left({x\_m}^{2}, {x\_m}^{2} \cdot \left(0.15625 \cdot t\_1 - \mathsf{fma}\left(0.125, \frac{\sqrt{0.5} \cdot \left(0.375 - 0.0625 \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{t\_2}, 0.25 \cdot \frac{\sqrt{0.5} \cdot t\_3}{\sqrt{2} \cdot t\_0}\right)\right) - t\_3, 0.25 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}{1 + \sqrt{\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (/ 1.0 t_0))
        (t_2 (* (sqrt 2.0) (pow t_0 2.0)))
        (t_3 (fma -0.0625 (/ (sqrt 0.5) t_2) (* 0.1875 t_1))))
   (if (<= x_m 0.008)
     (*
      (pow x_m 2.0)
      (fma
       (pow x_m 2.0)
       (-
        (*
         (pow x_m 2.0)
         (-
          (* 0.15625 t_1)
          (fma
           0.125
           (/
            (* (sqrt 0.5) (- 0.375 (* 0.0625 (/ 1.0 (pow (sqrt 2.0) 2.0)))))
            t_2)
           (* 0.25 (/ (* (sqrt 0.5) t_3) (* (sqrt 2.0) t_0))))))
        t_3)
       (* 0.25 t_1)))
     (/
      (- 1.0 (* (+ (cos (atan x_m)) 1.0) 0.5))
      (+ 1.0 (sqrt (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = 1.0 / t_0;
	double t_2 = sqrt(2.0) * pow(t_0, 2.0);
	double t_3 = fma(-0.0625, (sqrt(0.5) / t_2), (0.1875 * t_1));
	double tmp;
	if (x_m <= 0.008) {
		tmp = pow(x_m, 2.0) * fma(pow(x_m, 2.0), ((pow(x_m, 2.0) * ((0.15625 * t_1) - fma(0.125, ((sqrt(0.5) * (0.375 - (0.0625 * (1.0 / pow(sqrt(2.0), 2.0))))) / t_2), (0.25 * ((sqrt(0.5) * t_3) / (sqrt(2.0) * t_0)))))) - t_3), (0.25 * t_1));
	} else {
		tmp = (1.0 - ((cos(atan(x_m)) + 1.0) * 0.5)) / (1.0 + sqrt(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(sqrt(2.0) * (t_0 ^ 2.0))
	t_3 = fma(-0.0625, Float64(sqrt(0.5) / t_2), Float64(0.1875 * t_1))
	tmp = 0.0
	if (x_m <= 0.008)
		tmp = Float64((x_m ^ 2.0) * fma((x_m ^ 2.0), Float64(Float64((x_m ^ 2.0) * Float64(Float64(0.15625 * t_1) - fma(0.125, Float64(Float64(sqrt(0.5) * Float64(0.375 - Float64(0.0625 * Float64(1.0 / (sqrt(2.0) ^ 2.0))))) / t_2), Float64(0.25 * Float64(Float64(sqrt(0.5) * t_3) / Float64(sqrt(2.0) * t_0)))))) - t_3), Float64(0.25 * t_1)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$2), $MachinePrecision] + N[(0.1875 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.008], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.15625 * t$95$1), $MachinePrecision] - N[(0.125 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.375 - N[(0.0625 * N[(1.0 / N[Power[N[Sqrt[2.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(0.25 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * t$95$3), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0080000000000000002

   1. Initial program 64.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 27.2

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

   5. Applied rewrites 27.2%

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

      1. metadata-eval 27.2

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

      2. cos-atan-rev 27.2

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

      3. cos-atan-rev 27.2

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 72.9%

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

   if 0.0080000000000000002 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_2 := \frac{1}{t\_1}\\ t_3 := 1 + t\_0\\ \mathbf{if}\;x\_m \leq 0.0026:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {t\_1}^{2}}, 0.1875 \cdot t\_2\right), 0.25 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \sqrt{{\left(0.5 + t\_0 \cdot 0.5\right)}^{6}}}{1 + \mathsf{fma}\left(0.25, {t\_3}^{2}, 0.5 \cdot t\_3\right)}}{1 + \sqrt{\left(t\_0 + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m)))
        (t_1 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_2 (/ 1.0 t_1))
        (t_3 (+ 1.0 t_0)))
   (if (<= x_m 0.0026)
     (*
      (pow x_m 2.0)
      (fma
       -1.0
       (*
        (pow x_m 2.0)
        (fma
         -0.0625
         (/ (sqrt 0.5) (* (sqrt 2.0) (pow t_1 2.0)))
         (* 0.1875 t_2)))
       (* 0.25 t_2)))
     (/
      (/
       (- 1.0 (sqrt (pow (+ 0.5 (* t_0 0.5)) 6.0)))
       (+ 1.0 (fma 0.25 (pow t_3 2.0) (* 0.5 t_3))))
      (+ 1.0 (sqrt (* (+ t_0 1.0) 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_2 = 1.0 / t_1;
	double t_3 = 1.0 + t_0;
	double tmp;
	if (x_m <= 0.0026) {
		tmp = pow(x_m, 2.0) * fma(-1.0, (pow(x_m, 2.0) * fma(-0.0625, (sqrt(0.5) / (sqrt(2.0) * pow(t_1, 2.0))), (0.1875 * t_2))), (0.25 * t_2));
	} else {
		tmp = ((1.0 - sqrt(pow((0.5 + (t_0 * 0.5)), 6.0))) / (1.0 + fma(0.25, pow(t_3, 2.0), (0.5 * t_3)))) / (1.0 + sqrt(((t_0 + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_2 = Float64(1.0 / t_1)
	t_3 = Float64(1.0 + t_0)
	tmp = 0.0
	if (x_m <= 0.0026)
		tmp = Float64((x_m ^ 2.0) * fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.0625, Float64(sqrt(0.5) / Float64(sqrt(2.0) * (t_1 ^ 2.0))), Float64(0.1875 * t_2))), Float64(0.25 * t_2)));
	else
		tmp = Float64(Float64(Float64(1.0 - sqrt((Float64(0.5 + Float64(t_0 * 0.5)) ^ 6.0))) / Float64(1.0 + fma(0.25, (t_3 ^ 2.0), Float64(0.5 * t_3)))) / Float64(1.0 + sqrt(Float64(Float64(t_0 + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0026], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Sqrt[N[Power[N[(0.5 + N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision], 6.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.25 * N[Power[t$95$3, 2.0], $MachinePrecision] + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0025999999999999999

   1. Initial program 64.3%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. metadata-eval 27.5

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

      2. cos-atan-rev 27.5

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

      3. cos-atan-rev 27.5

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 71.7%

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

   if 0.0025999999999999999 < x

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-atan.f64 99.4

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

   9. Applied rewrites 99.4%

      \[\leadsto \frac{\frac{1 - {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{3}}{1 + \color{blue}{\mathsf{fma}\left(0.25, {\left(1 + \cos \tan^{-1} x\right)}^{2}, 0.5 \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

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

       2. sqrt-unprod N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lift-pow.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. pow-prod-up N/A

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

       7. metadata-eval N/A

          \[\leadsto \frac{\frac{1 - \sqrt{{\left(\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}\right)}^{\color{blue}{6}}}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

       8. lower-pow.f64 99.5

          \[\leadsto \frac{\frac{1 - \sqrt{\color{blue}{{\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{6}}}}{1 + \mathsf{fma}\left(0.25, {\left(1 + \cos \tan^{-1} x\right)}^{2}, 0.5 \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   11. Applied rewrites 99.5%

       \[\leadsto \frac{\frac{1 - \color{blue}{\sqrt{{\left(0.5 + \cos \tan^{-1} x \cdot 0.5\right)}^{6}}}}{1 + \mathsf{fma}\left(0.25, {\left(1 + \cos \tan^{-1} x\right)}^{2}, 0.5 \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \cos \tan^{-1} x\_m\\ t_2 := \frac{1}{t\_0}\\ t_3 := 1 + t\_1\\ \mathbf{if}\;x\_m \leq 0.0026:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {t\_0}^{2}}, 0.1875 \cdot t\_2\right), 0.25 \cdot t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {\left(\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5\right)}^{3}}{1 + \mathsf{fma}\left(0.25, {t\_3}^{2}, 0.5 \cdot t\_3\right)}}{1 + \sqrt{\left(t\_1 + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0))))
        (t_1 (cos (atan x_m)))
        (t_2 (/ 1.0 t_0))
        (t_3 (+ 1.0 t_1)))
   (if (<= x_m 0.0026)
     (*
      (pow x_m 2.0)
      (fma
       -1.0
       (*
        (pow x_m 2.0)
        (fma
         -0.0625
         (/ (sqrt 0.5) (* (sqrt 2.0) (pow t_0 2.0)))
         (* 0.1875 t_2)))
       (* 0.25 t_2)))
     (/
      (/
       (- 1.0 (pow (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5) 3.0))
       (+ 1.0 (fma 0.25 (pow t_3 2.0) (* 0.5 t_3))))
      (+ 1.0 (sqrt (* (+ t_1 1.0) 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = cos(atan(x_m));
	double t_2 = 1.0 / t_0;
	double t_3 = 1.0 + t_1;
	double tmp;
	if (x_m <= 0.0026) {
		tmp = pow(x_m, 2.0) * fma(-1.0, (pow(x_m, 2.0) * fma(-0.0625, (sqrt(0.5) / (sqrt(2.0) * pow(t_0, 2.0))), (0.1875 * t_2))), (0.25 * t_2));
	} else {
		tmp = ((1.0 - pow(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5), 3.0)) / (1.0 + fma(0.25, pow(t_3, 2.0), (0.5 * t_3)))) / (1.0 + sqrt(((t_1 + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = cos(atan(x_m))
	t_2 = Float64(1.0 / t_0)
	t_3 = Float64(1.0 + t_1)
	tmp = 0.0
	if (x_m <= 0.0026)
		tmp = Float64((x_m ^ 2.0) * fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.0625, Float64(sqrt(0.5) / Float64(sqrt(2.0) * (t_0 ^ 2.0))), Float64(0.1875 * t_2))), Float64(0.25 * t_2)));
	else
		tmp = Float64(Float64(Float64(1.0 - (Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5) ^ 3.0)) / Float64(1.0 + fma(0.25, (t_3 ^ 2.0), Float64(0.5 * t_3)))) / Float64(1.0 + sqrt(Float64(Float64(t_1 + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0026], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Power[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.25 * N[Power[t$95$3, 2.0], $MachinePrecision] + N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(t$95$1 + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0025999999999999999

   1. Initial program 64.3%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. metadata-eval 27.5

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

      2. cos-atan-rev 27.5

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

      3. cos-atan-rev 27.5

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 71.7%

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

   if 0.0025999999999999999 < x

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-atan.f64 99.4

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

   9. Applied rewrites 99.4%

      \[\leadsto \frac{\frac{1 - {\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{3}}{1 + \color{blue}{\mathsf{fma}\left(0.25, {\left(1 + \cos \tan^{-1} x\right)}^{2}, 0.5 \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
   10. Step-by-step derivation

       1. lift-atan.f64 N/A

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

       2. lift-cos.f64 N/A

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

       3. cos-atan-rev N/A

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

       4. metadata-eval N/A

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

       5. pow2 N/A

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

       6. +-commutative N/A

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

       7. pow2 N/A

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

       8. sqrt-undiv N/A

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

       9. lower-sqrt.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lift-fma.f64 99.4

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

   11. Applied rewrites 99.4%

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

Alternative 5: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \sqrt{0.5} \cdot \sqrt{2}\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;x\_m \leq 0.0026:\\ \;\;\;\;{x\_m}^{2} \cdot \mathsf{fma}\left(-1, {x\_m}^{2} \cdot \mathsf{fma}\left(-0.0625, \frac{\sqrt{0.5}}{\sqrt{2} \cdot {t\_0}^{2}}, 0.1875 \cdot t\_1\right), 0.25 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}{1 + \sqrt{\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (sqrt 0.5) (sqrt 2.0)))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 0.0026)
     (*
      (pow x_m 2.0)
      (fma
       -1.0
       (*
        (pow x_m 2.0)
        (fma
         -0.0625
         (/ (sqrt 0.5) (* (sqrt 2.0) (pow t_0 2.0)))
         (* 0.1875 t_1)))
       (* 0.25 t_1)))
     (/
      (- 1.0 (* (+ (cos (atan x_m)) 1.0) 0.5))
      (+ 1.0 (sqrt (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (sqrt(0.5) * sqrt(2.0));
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 0.0026) {
		tmp = pow(x_m, 2.0) * fma(-1.0, (pow(x_m, 2.0) * fma(-0.0625, (sqrt(0.5) / (sqrt(2.0) * pow(t_0, 2.0))), (0.1875 * t_1))), (0.25 * t_1));
	} else {
		tmp = (1.0 - ((cos(atan(x_m)) + 1.0) * 0.5)) / (1.0 + sqrt(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 0.0026)
		tmp = Float64((x_m ^ 2.0) * fma(-1.0, Float64((x_m ^ 2.0) * fma(-0.0625, Float64(sqrt(0.5) / Float64(sqrt(2.0) * (t_0 ^ 2.0))), Float64(0.1875 * t_1))), Float64(0.25 * t_1)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0026], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(-0.0625 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.1875 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0025999999999999999

   1. Initial program 64.3%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 27.5

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

   5. Applied rewrites 27.5%

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

      1. metadata-eval 27.5

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

      2. cos-atan-rev 27.5

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

      3. cos-atan-rev 27.5

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 71.7%

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

   if 0.0025999999999999999 < x

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 6: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000108:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}{1 + \sqrt{\left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000108)
   (* 0.125 (* x_m x_m))
   (/
    (- 1.0 (* (+ (cos (atan x_m)) 1.0) 0.5))
    (+ 1.0 (sqrt (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000108) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = (1.0 - ((cos(atan(x_m)) + 1.0) * 0.5)) / (1.0 + sqrt(((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000108)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(sqrt(Float64(1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000108], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Sqrt[N[(1.0 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.08e-4

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 72.3

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

   8. Applied rewrites 72.3%

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

   if 1.08e-4 < x

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. metadata-eval N/A

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

      5. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 7: 98.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00017:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00017)
   (* 0.125 (* x_m x_m))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00017) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00017)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00017], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e-4

   1. Initial program 64.3%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 72.3

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

   8. Applied rewrites 72.3%

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

   if 1.7e-4 < x

   1. Initial program 97.8%

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

      1. lift-hypot.f64 N/A

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 8: 98.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.25) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 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 <= 1.25d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt(((0.5d0 / x_m) + 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 <= 1.25) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - Math.sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.25:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt(((0.5 / x_m) + 0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.25], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 64.4%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 72.0

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

   8. Applied rewrites 72.0%

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

   if 1.25 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 9: 98.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 0.5 / (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 <= 1.5d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 0.5d0 / (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 <= 1.5) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(0.5 / 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 <= 1.5)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 0.5 / (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, 1.5], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 64.4%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 72.0

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

   8. Applied rewrites 72.0%

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

   if 1.5 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. metadata-eval 98.5

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

      2. cos-atan-rev 98.5

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

      3. cos-atan-rev 98.5

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 97.9

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

   10. Applied rewrites 97.9%

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

Alternative 10: 97.6% accurate, 6.7× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.5d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.5:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.5)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 64.4%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 37.5

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 72.0

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

   8. Applied rewrites 72.0%

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

   if 1.5 < 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 96.4%

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

Alternative 11: 51.4% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 0.125d0 * (x_m * x_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

3. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. sqrt-undiv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 30.0

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

5. Applied rewrites 30.0%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 56.6

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

8. Applied rewrites 56.6%

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

Alternative 12: 27.4% accurate, 134.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.1%

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

3. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 28.4

      \[\leadsto 0 \]

5. Applied rewrites 28.4%

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

Reproduce

herbie shell --seed 2025066 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))