Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.9%

Time: 5.8s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\\ t_1 := {t\_0}^{2}\\ t_2 := t\_0 \cdot 3\\ t_3 := \frac{0.75}{t\_2}\\ t_4 := \mathsf{fma}\left(t\_0, -0.375, \frac{-3 \cdot \left({0.5}^{1.5} \cdot 3\right)}{\sqrt{8}}\right)\\ t_5 := \frac{t\_4}{t\_1} \cdot 0\\ \mathbf{if}\;x\_m \leq 0.0025:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, \frac{-0.75}{t\_2} - \mathsf{fma}\left(\frac{\mathsf{fma}\left(-3, {0.5}^{1.5} \cdot \frac{-0.375}{\sqrt{8}}, \mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.5, \sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.1875\right), \frac{0.5 \cdot \left({0.5}^{1.5} \cdot 14.625\right)}{\sqrt{8}}\right)\right)}{t\_1}, 0, \frac{t\_4}{t\_0} \cdot \frac{t\_3 - t\_5}{3}\right), t\_3\right) - t\_5, \frac{\frac{0}{t\_0}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\_m\right) \cdot 0.5} \cdot \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (pow 0.5 1.5) (sqrt 8.0) 1.0))
        (t_1 (pow t_0 2.0))
        (t_2 (* t_0 3.0))
        (t_3 (/ 0.75 t_2))
        (t_4 (fma t_0 -0.375 (/ (* -3.0 (* (pow 0.5 1.5) 3.0)) (sqrt 8.0))))
        (t_5 (* (/ t_4 t_1) 0.0)))
   (if (<= x_m 0.0025)
     (fma
      (* x_m x_m)
      (-
       (fma
        (* x_m x_m)
        (-
         (/ -0.75 t_2)
         (fma
          (/
           (fma
            -3.0
            (* (pow 0.5 1.5) (/ -0.375 (sqrt 8.0)))
            (fma
             t_0
             (fma 0.5 (* (sqrt 0.5) (/ 0.34375 (sqrt 2.0))) 0.1875)
             (/ (* 0.5 (* (pow 0.5 1.5) 14.625)) (sqrt 8.0))))
           t_1)
          0.0
          (* (/ t_4 t_0) (/ (- t_3 t_5) 3.0))))
        t_3)
       t_5)
      (/ (/ 0.0 t_0) 3.0))
     (/
      (- 1.0 (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))
      (+ 1.0 (* (exp (* (log1p (cos (atan x_m))) 0.5)) (sqrt 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(pow(0.5, 1.5), sqrt(8.0), 1.0);
	double t_1 = pow(t_0, 2.0);
	double t_2 = t_0 * 3.0;
	double t_3 = 0.75 / t_2;
	double t_4 = fma(t_0, -0.375, ((-3.0 * (pow(0.5, 1.5) * 3.0)) / sqrt(8.0)));
	double t_5 = (t_4 / t_1) * 0.0;
	double tmp;
	if (x_m <= 0.0025) {
		tmp = fma((x_m * x_m), (fma((x_m * x_m), ((-0.75 / t_2) - fma((fma(-3.0, (pow(0.5, 1.5) * (-0.375 / sqrt(8.0))), fma(t_0, fma(0.5, (sqrt(0.5) * (0.34375 / sqrt(2.0))), 0.1875), ((0.5 * (pow(0.5, 1.5) * 14.625)) / sqrt(8.0)))) / t_1), 0.0, ((t_4 / t_0) * ((t_3 - t_5) / 3.0)))), t_3) - t_5), ((0.0 / t_0) / 3.0));
	} else {
		tmp = (1.0 - (((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / (1.0 + (exp((log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma((0.5 ^ 1.5), sqrt(8.0), 1.0)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(t_0 * 3.0)
	t_3 = Float64(0.75 / t_2)
	t_4 = fma(t_0, -0.375, Float64(Float64(-3.0 * Float64((0.5 ^ 1.5) * 3.0)) / sqrt(8.0)))
	t_5 = Float64(Float64(t_4 / t_1) * 0.0)
	tmp = 0.0
	if (x_m <= 0.0025)
		tmp = fma(Float64(x_m * x_m), Float64(fma(Float64(x_m * x_m), Float64(Float64(-0.75 / t_2) - fma(Float64(fma(-3.0, Float64((0.5 ^ 1.5) * Float64(-0.375 / sqrt(8.0))), fma(t_0, fma(0.5, Float64(sqrt(0.5) * Float64(0.34375 / sqrt(2.0))), 0.1875), Float64(Float64(0.5 * Float64((0.5 ^ 1.5) * 14.625)) / sqrt(8.0)))) / t_1), 0.0, Float64(Float64(t_4 / t_0) * Float64(Float64(t_3 - t_5) / 3.0)))), t_3) - t_5), Float64(Float64(0.0 / t_0) / 3.0));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / Float64(1.0 + Float64(exp(Float64(log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Power[0.5, 1.5], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * 3.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.75 / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * -0.375 + N[(N[(-3.0 * N[(N[Power[0.5, 1.5], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 / t$95$1), $MachinePrecision] * 0.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0025], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(-0.75 / t$95$2), $MachinePrecision] - N[(N[(N[(-3.0 * N[(N[Power[0.5, 1.5], $MachinePrecision] * N[(-0.375 / N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(0.5 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(0.34375 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.1875), $MachinePrecision] + N[(N[(0.5 * N[(N[Power[0.5, 1.5], $MachinePrecision] * 14.625), $MachinePrecision]), $MachinePrecision] / N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * 0.0 + N[(N[(t$95$4 / t$95$0), $MachinePrecision] * N[(N[(t$95$3 - t$95$5), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] - t$95$5), $MachinePrecision] + N[(N[(0.0 / t$95$0), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Exp[N[(N[Log[1 + N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00250000000000000005

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sqrt-undiv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 34.7

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

   5. Applied rewrites 34.7%

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

   7. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{3}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}} \]

   9. Applied rewrites 33.6%

      \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{6}}{1 + {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{3}}}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\left(6 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)} + {x}^{2} \cdot \left(-6 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)} - \left(\frac{\left(-3 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}{\sqrt{8}} + \left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(\frac{-1}{2} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right)\right) \cdot \left(6 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)} - \frac{\left(-3 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}{\sqrt{8}} + \left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(\frac{-1}{2} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right)\right) \cdot \left(1 - 8 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{6}\right)}{{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right)}^{2} \cdot {\left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}^{2}}\right)}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)} + \frac{\left(-3 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(\frac{-1}{2} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right)}{\sqrt{8}} + \left(\frac{1}{2} \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(\left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(6 - 9 \cdot \frac{1}{{\left(\sqrt{8}\right)}^{2}}\right)\right)}{\sqrt{8}} + \left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(\frac{3}{8} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \frac{1}{2} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2}}\right)\right)\right) \cdot \left(1 - 8 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{6}\right)}{{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right)}^{2} \cdot {\left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}^{2}}\right)\right)\right) - \frac{\left(-3 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}{\sqrt{8}} + \left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(\frac{-1}{2} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right)\right) \cdot \left(1 - 8 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{6}\right)}{{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right)}^{2} \cdot {\left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}^{2}}\right) + \frac{1}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right) - 8 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}} \]

   12. Applied rewrites 68.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{-0.75}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right) \cdot 3} - \mathsf{fma}\left(\frac{\mathsf{fma}\left(-3, {0.5}^{1.5} \cdot \frac{-0.375}{\sqrt{8}}, \mathsf{fma}\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right), \mathsf{fma}\left(0.5, \sqrt{0.5} \cdot \frac{0.34375}{\sqrt{2}}, 0.1875\right), \frac{0.5 \cdot \left({0.5}^{1.5} \cdot 14.625\right)}{\sqrt{8}}\right)\right)}{{\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\right)}^{2}}, 0, \frac{\mathsf{fma}\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right), -0.375, \frac{-3 \cdot \left({0.5}^{1.5} \cdot 3\right)}{\sqrt{8}}\right)}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)} \cdot \frac{\frac{0.75}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right) \cdot 3} - \frac{\mathsf{fma}\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right), -0.375, \frac{-3 \cdot \left({0.5}^{1.5} \cdot 3\right)}{\sqrt{8}}\right)}{{\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\right)}^{2}} \cdot 0}{3}\right), \frac{0.75}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right) \cdot 3}\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right), -0.375, \frac{-3 \cdot \left({0.5}^{1.5} \cdot 3\right)}{\sqrt{8}}\right)}{{\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\right)}^{2}} \cdot 0, \frac{\frac{0}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)}}{3}\right)} \]

   if 0.00250000000000000005 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sqrt-prod N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 N/A

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\color{blue}{\mathsf{log1p}\left(\cos \tan^{-1} x\right)} \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\color{blue}{\cos \tan^{-1} x}\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      12. lift-atan.f64 100.0

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \color{blue}{\tan^{-1} x}\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \color{blue}{e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5}} \cdot \sqrt{0.5}} \]
   9. Step-by-step derivation

      1. lift-atan.f64 N/A

         \[\leadsto \frac{1 - \left(\cos \color{blue}{\tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\cos \tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      3. cos-atan-rev N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\color{blue}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{x \cdot x} + 1}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      9. lower-fma.f64 100.0

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\\ \mathbf{if}\;x\_m \leq 0.000145:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \frac{0.75}{t\_0 \cdot 3} - \frac{\mathsf{fma}\left(t\_0, -0.375, \frac{-3 \cdot \left({0.5}^{1.5} \cdot 3\right)}{\sqrt{8}}\right)}{{t\_0}^{2}} \cdot 0, \frac{\frac{0}{t\_0}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\_m\right) \cdot 0.5} \cdot \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (pow 0.5 1.5) (sqrt 8.0) 1.0)))
   (if (<= x_m 0.000145)
     (fma
      (* x_m x_m)
      (-
       (/ 0.75 (* t_0 3.0))
       (*
        (/
         (fma t_0 -0.375 (/ (* -3.0 (* (pow 0.5 1.5) 3.0)) (sqrt 8.0)))
         (pow t_0 2.0))
        0.0))
      (/ (/ 0.0 t_0) 3.0))
     (/
      (- 1.0 (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))
      (+ 1.0 (* (exp (* (log1p (cos (atan x_m))) 0.5)) (sqrt 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(pow(0.5, 1.5), sqrt(8.0), 1.0);
	double tmp;
	if (x_m <= 0.000145) {
		tmp = fma((x_m * x_m), ((0.75 / (t_0 * 3.0)) - ((fma(t_0, -0.375, ((-3.0 * (pow(0.5, 1.5) * 3.0)) / sqrt(8.0))) / pow(t_0, 2.0)) * 0.0)), ((0.0 / t_0) / 3.0));
	} else {
		tmp = (1.0 - (((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / (1.0 + (exp((log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma((0.5 ^ 1.5), sqrt(8.0), 1.0)
	tmp = 0.0
	if (x_m <= 0.000145)
		tmp = fma(Float64(x_m * x_m), Float64(Float64(0.75 / Float64(t_0 * 3.0)) - Float64(Float64(fma(t_0, -0.375, Float64(Float64(-3.0 * Float64((0.5 ^ 1.5) * 3.0)) / sqrt(8.0))) / (t_0 ^ 2.0)) * 0.0)), Float64(Float64(0.0 / t_0) / 3.0));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / Float64(1.0 + Float64(exp(Float64(log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Power[0.5, 1.5], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.000145], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.75 / N[(t$95$0 * 3.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t$95$0 * -0.375 + N[(N[(-3.0 * N[(N[Power[0.5, 1.5], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.0 / t$95$0), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Exp[N[(N[Log[1 + N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-4

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sqrt-undiv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 34.7

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

   5. Applied rewrites 34.7%

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

   7. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{3}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}} \]

   9. Applied rewrites 33.6%

      \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{6}}{1 + {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{3}}}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(6 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)} - \frac{\left(-3 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}{\sqrt{8}} + \left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(\frac{-1}{2} \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}}\right)\right) \cdot \left(1 - 8 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{6}\right)}{{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right)}^{2} \cdot {\left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}^{2}}\right) + \frac{1}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}\right) - 8 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right) \cdot \left(1 + \left(2 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{2} + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)\right)}} \]

   12. Applied rewrites 69.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.75}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right) \cdot 3} - \frac{\mathsf{fma}\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right), -0.375, \frac{-3 \cdot \left({0.5}^{1.5} \cdot 3\right)}{\sqrt{8}}\right)}{{\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\right)}^{2}} \cdot 0, \frac{\frac{0}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)}}{3}\right)} \]

   if 1.45e-4 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sqrt-prod N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 N/A

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\color{blue}{\mathsf{log1p}\left(\cos \tan^{-1} x\right)} \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\color{blue}{\cos \tan^{-1} x}\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      12. lift-atan.f64 100.0

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \color{blue}{\tan^{-1} x}\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \color{blue}{e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5}} \cdot \sqrt{0.5}} \]
   9. Step-by-step derivation

      1. lift-atan.f64 N/A

         \[\leadsto \frac{1 - \left(\cos \color{blue}{\tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\cos \tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      3. cos-atan-rev N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\color{blue}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{x \cdot x} + 1}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      9. lower-fma.f64 100.0

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\\ t_1 := \mathsf{fma}\left(-0.125, x\_m \cdot x\_m, 1\right)\\ \mathbf{if}\;x\_m \leq 0.000145:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, \frac{0.75}{t\_0} - \frac{-3 \cdot \left({0.5}^{1.5} \cdot 0\right)}{\sqrt{8} \cdot {t\_0}^{2}}, \frac{0}{t\_0}\right)}{1 + \left({t\_1}^{2} + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\_m\right) \cdot 0.5} \cdot \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (pow 0.5 1.5) (sqrt 8.0) 1.0))
        (t_1 (fma -0.125 (* x_m x_m) 1.0)))
   (if (<= x_m 0.000145)
     (/
      (fma
       (* x_m x_m)
       (-
        (/ 0.75 t_0)
        (/ (* -3.0 (* (pow 0.5 1.5) 0.0)) (* (sqrt 8.0) (pow t_0 2.0))))
       (/ 0.0 t_0))
      (+ 1.0 (+ (pow t_1 2.0) t_1)))
     (/
      (- 1.0 (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))
      (+ 1.0 (* (exp (* (log1p (cos (atan x_m))) 0.5)) (sqrt 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(pow(0.5, 1.5), sqrt(8.0), 1.0);
	double t_1 = fma(-0.125, (x_m * x_m), 1.0);
	double tmp;
	if (x_m <= 0.000145) {
		tmp = fma((x_m * x_m), ((0.75 / t_0) - ((-3.0 * (pow(0.5, 1.5) * 0.0)) / (sqrt(8.0) * pow(t_0, 2.0)))), (0.0 / t_0)) / (1.0 + (pow(t_1, 2.0) + t_1));
	} else {
		tmp = (1.0 - (((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / (1.0 + (exp((log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma((0.5 ^ 1.5), sqrt(8.0), 1.0)
	t_1 = fma(-0.125, Float64(x_m * x_m), 1.0)
	tmp = 0.0
	if (x_m <= 0.000145)
		tmp = Float64(fma(Float64(x_m * x_m), Float64(Float64(0.75 / t_0) - Float64(Float64(-3.0 * Float64((0.5 ^ 1.5) * 0.0)) / Float64(sqrt(8.0) * (t_0 ^ 2.0)))), Float64(0.0 / t_0)) / Float64(1.0 + Float64((t_1 ^ 2.0) + t_1)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / Float64(1.0 + Float64(exp(Float64(log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Power[0.5, 1.5], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(-0.125 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.000145], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.75 / t$95$0), $MachinePrecision] - N[(N[(-3.0 * N[(N[Power[0.5, 1.5], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Exp[N[(N[Log[1 + N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-4

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sqrt-undiv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 34.7

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

   5. Applied rewrites 34.7%

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

   7. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{3}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}} \]

   9. Applied rewrites 33.6%

      \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{6}}{1 + {\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{3}}}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(6 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}} - -3 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \left(1 - 8 \cdot {\left(\sqrt{\frac{1}{2}}\right)}^{6}\right)}{\sqrt{8} \cdot {\left(1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}\right)}^{2}}\right) + \frac{1}{1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}}\right) - 8 \cdot \frac{{\left(\sqrt{\frac{1}{2}}\right)}^{6}}{1 + {\left(\sqrt{\frac{1}{2}}\right)}^{3} \cdot \sqrt{8}}}}{1 + \left({\left(\mathsf{fma}\left(\frac{-1}{8}, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(\frac{-1}{8}, x \cdot x, 1\right)\right)} \]

   12. Applied rewrites 68.9%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \frac{0.75}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)} - \frac{-3 \cdot \left({0.5}^{1.5} \cdot 0\right)}{\sqrt{8} \cdot {\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\right)}^{2}}, \frac{0}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)}\right)}}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + 1 \cdot \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)} \]

   if 1.45e-4 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sqrt-prod N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 N/A

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\color{blue}{\mathsf{log1p}\left(\cos \tan^{-1} x\right)} \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\color{blue}{\cos \tan^{-1} x}\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      12. lift-atan.f64 100.0

          \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \color{blue}{\tan^{-1} x}\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \color{blue}{e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5}} \cdot \sqrt{0.5}} \]
   9. Step-by-step derivation

      1. lift-atan.f64 N/A

         \[\leadsto \frac{1 - \left(\cos \color{blue}{\tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      2. lift-cos.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\cos \tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      3. cos-atan-rev N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\color{blue}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      6. pow2 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      7. +-commutative N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{x \cdot x} + 1}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

      9. lower-fma.f64 100.0

         \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000145:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \frac{0.75}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)} - \frac{-3 \cdot \left({0.5}^{1.5} \cdot 0\right)}{\sqrt{8} \cdot {\left(\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)\right)}^{2}}, \frac{0}{\mathsf{fma}\left({0.5}^{1.5}, \sqrt{8}, 1\right)}\right)}{1 + \left({\left(\mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}^{2} + \mathsf{fma}\left(-0.125, x \cdot x, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1 - \left(\frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\_m\right) \cdot 0.5} \cdot \sqrt{0.5}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/
  (- 1.0 (* (+ (/ 1.0 (sqrt (fma x_m x_m 1.0))) 1.0) 0.5))
  (+ 1.0 (* (exp (* (log1p (cos (atan x_m))) 0.5)) (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	return (1.0 - (((1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / (1.0 + (exp((log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(1.0 - Float64(Float64(Float64(1.0 / sqrt(fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / Float64(1.0 + Float64(exp(Float64(log1p(cos(atan(x_m))) * 0.5)) * sqrt(0.5))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(1.0 - N[(N[(N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[Exp[N[(N[Log[1 + N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 75.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. sqrt-prod N/A

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

   7. +-commutative N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 75.2%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-log1p.f64 N/A

       \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\color{blue}{\mathsf{log1p}\left(\cos \tan^{-1} x\right)} \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   11. lift-cos.f64 N/A

       \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\color{blue}{\cos \tan^{-1} x}\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   12. lift-atan.f64 75.2

       \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \color{blue}{\tan^{-1} x}\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

8. Applied rewrites 75.2%

   \[\leadsto \frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \color{blue}{e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5}} \cdot \sqrt{0.5}} \]
9. Step-by-step derivation

   1. lift-atan.f64 N/A

      \[\leadsto \frac{1 - \left(\cos \color{blue}{\tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   2. lift-cos.f64 N/A

      \[\leadsto \frac{1 - \left(\color{blue}{\cos \tan^{-1} x} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   3. cos-atan-rev N/A

      \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{1 - \left(\frac{1}{\color{blue}{\sqrt{1 + x \cdot x}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   6. pow2 N/A

      \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{1 + \color{blue}{{x}^{2}}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{{x}^{2} + 1}}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   8. pow2 N/A

      \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{x \cdot x} + 1}} + 1\right) \cdot \frac{1}{2}}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot \frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} \]

   9. lower-fma.f64 75.2

      \[\leadsto \frac{1 - \left(\frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]

10. Applied rewrites 75.2%

    \[\leadsto \frac{1 - \left(\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} + 1\right) \cdot 0.5}{1 + e^{\mathsf{log1p}\left(\cos \tan^{-1} x\right) \cdot 0.5} \cdot \sqrt{0.5}} \]
11. Add Preprocessing

Alternative 5: 76.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 75.2%

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

   1. lift-atan.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-atan-rev N/A

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

   4. metadata-eval N/A

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

   5. pow2 N/A

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

   6. +-commutative N/A

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

   7. pow2 N/A

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

   8. sqrt-undiv N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-fma.f64 75.2

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

6. Applied rewrites 75.2%

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

Alternative 6: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 5 \cdot 10^{-15}:\\ \;\;\;\;1 - \mathsf{fma}\left(x\_m \cdot x\_m, -0.125, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m)))))) 5e-15)
   (- 1.0 (fma (* x_m x_m) -0.125 1.0))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))))) <= 5e-15) {
		tmp = 1.0 - fma((x_m * x_m), -0.125, 1.0);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m)))))) <= 5e-15)
		tmp = Float64(1.0 - fma(Float64(x_m * x_m), -0.125, 1.0));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-15], N[(1.0 - N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))) < 4.99999999999999999e-15

   1. Initial program 48.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sqrt-undiv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 48.9

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

   5. Applied rewrites 48.9%

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.5, -0.25, 1\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. sqrt-undiv N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-undiv N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 48.9

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

   7. Applied rewrites 48.9%

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

   if 4.99999999999999999e-15 < (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))))

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. 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 97.9

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

   5. Applied rewrites 97.9%

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

Alternative 7: 75.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.4%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 74.4

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

4. Applied rewrites 74.4%

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

Alternative 8: 74.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 73.7%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 38.8

         \[\leadsto 0 \]

   5. Applied rewrites 38.8%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 75.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 74.1%

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

Alternative 9: 26.8% accurate, 134.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
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 25.1

      \[\leadsto 0 \]

5. Applied rewrites 25.1%

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))