Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.9%

Time: 4.5s

Alternatives: 12

Speedup: 6.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \cos \tan^{-1} x\_m\\ t_1 := 1 + t\_0\\ t_2 := t\_0 + 1\\ t_3 := \mathsf{fma}\left(\sqrt{t\_2}, \sqrt{0.5}, 1\right)\\ \mathbf{if}\;x\_m \leq 0.027:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, \frac{x\_m \cdot x\_m}{t\_3}, \frac{0.15625}{t\_3}\right) \cdot \left(x\_m \cdot x\_m\right) - \frac{0.1875}{t\_3}, x\_m \cdot x\_m, \frac{0.25}{t\_3}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - {t\_2}^{3} \cdot 0.125}{1 + \mathsf{fma}\left(0.25, {t\_1}^{2}, 0.5 \cdot t\_1\right)}}{1 + \sqrt{t\_2 \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (cos (atan x_m)))
        (t_1 (+ 1.0 t_0))
        (t_2 (+ t_0 1.0))
        (t_3 (fma (sqrt t_2) (sqrt 0.5) 1.0)))
   (if (<= x_m 0.027)
     (*
      (fma
       (-
        (* (fma -0.13671875 (/ (* x_m x_m) t_3) (/ 0.15625 t_3)) (* x_m x_m))
        (/ 0.1875 t_3))
       (* x_m x_m)
       (/ 0.25 t_3))
      (* x_m x_m))
     (/
      (/
       (- 1.0 (* (pow t_2 3.0) 0.125))
       (+ 1.0 (fma 0.25 (pow t_1 2.0) (* 0.5 t_1))))
      (+ 1.0 (sqrt (* t_2 0.5)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = cos(atan(x_m));
	double t_1 = 1.0 + t_0;
	double t_2 = t_0 + 1.0;
	double t_3 = fma(sqrt(t_2), sqrt(0.5), 1.0);
	double tmp;
	if (x_m <= 0.027) {
		tmp = fma(((fma(-0.13671875, ((x_m * x_m) / t_3), (0.15625 / t_3)) * (x_m * x_m)) - (0.1875 / t_3)), (x_m * x_m), (0.25 / t_3)) * (x_m * x_m);
	} else {
		tmp = ((1.0 - (pow(t_2, 3.0) * 0.125)) / (1.0 + fma(0.25, pow(t_1, 2.0), (0.5 * t_1)))) / (1.0 + sqrt((t_2 * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = cos(atan(x_m))
	t_1 = Float64(1.0 + t_0)
	t_2 = Float64(t_0 + 1.0)
	t_3 = fma(sqrt(t_2), sqrt(0.5), 1.0)
	tmp = 0.0
	if (x_m <= 0.027)
		tmp = Float64(fma(Float64(Float64(fma(-0.13671875, Float64(Float64(x_m * x_m) / t_3), Float64(0.15625 / t_3)) * Float64(x_m * x_m)) - Float64(0.1875 / t_3)), Float64(x_m * x_m), Float64(0.25 / t_3)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64((t_2 ^ 3.0) * 0.125)) / Float64(1.0 + fma(0.25, (t_1 ^ 2.0), Float64(0.5 * t_1)))) / Float64(1.0 + sqrt(Float64(t_2 * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[t$95$2], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.027], N[(N[(N[(N[(N[(-0.13671875 * N[(N[(x$95$m * x$95$m), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.15625 / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(0.1875 / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.25 / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[Power[t$95$2, 3.0], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.25 * N[Power[t$95$1, 2.0], $MachinePrecision] + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(t$95$2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0269999999999999997

   1. Initial program 68.2%

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

      1. lift--.f64 N/A

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

      2. 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 68.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.2%

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

   if 0.0269999999999999997 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. metadata-eval N/A

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

      8. flip-- N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot 0.125}{1 + \left({\left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)}^{2} + 1 \cdot \left(\left(\cos \tan^{-1} x + 1\right) \cdot 0.5\right)\right)}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   7. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot \frac{1}{8}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot \frac{1}{8}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      5. lift-atan.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot \frac{1}{8}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot \frac{1}{8}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot \frac{1}{8}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      8. lift-cos.f64 N/A

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot \frac{1}{8}}{1 + \mathsf{fma}\left(\frac{1}{4}, {\left(1 + \cos \tan^{-1} x\right)}^{2}, \frac{1}{2} \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      9. lift-atan.f64 100.0

         \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot 0.125}{1 + \mathsf{fma}\left(0.25, {\left(1 + \cos \tan^{-1} x\right)}^{2}, 0.5 \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   9. Applied rewrites 100.0%

      \[\leadsto \frac{\frac{1 - {\left(\cos \tan^{-1} x + 1\right)}^{3} \cdot 0.125}{1 + \color{blue}{\mathsf{fma}\left(0.25, {\left(1 + \cos \tan^{-1} x\right)}^{2}, 0.5 \cdot \left(1 + \cos \tan^{-1} x\right)\right)}}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{\cos \tan^{-1} x\_m + 1}, \sqrt{0.5}, 1\right)\\ t_1 := \frac{0.5}{x\_m} + 0.5\\ \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, \frac{x\_m \cdot x\_m}{t\_0}, \frac{0.15625}{t\_0}\right) \cdot \left(x\_m \cdot x\_m\right) - \frac{0.1875}{t\_0}, x\_m \cdot x\_m, \frac{0.25}{t\_0}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{1 + \sqrt{t\_1}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (sqrt (+ (cos (atan x_m)) 1.0)) (sqrt 0.5) 1.0))
        (t_1 (+ (/ 0.5 x_m) 0.5)))
   (if (<= x_m 1.05)
     (*
      (fma
       (-
        (* (fma -0.13671875 (/ (* x_m x_m) t_0) (/ 0.15625 t_0)) (* x_m x_m))
        (/ 0.1875 t_0))
       (* x_m x_m)
       (/ 0.25 t_0))
      (* x_m x_m))
     (/ (- 1.0 t_1) (+ 1.0 (sqrt t_1))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(sqrt((cos(atan(x_m)) + 1.0)), sqrt(0.5), 1.0);
	double t_1 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.05) {
		tmp = fma(((fma(-0.13671875, ((x_m * x_m) / t_0), (0.15625 / t_0)) * (x_m * x_m)) - (0.1875 / t_0)), (x_m * x_m), (0.25 / t_0)) * (x_m * x_m);
	} else {
		tmp = (1.0 - t_1) / (1.0 + sqrt(t_1));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(sqrt(Float64(cos(atan(x_m)) + 1.0)), sqrt(0.5), 1.0)
	t_1 = Float64(Float64(0.5 / x_m) + 0.5)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(fma(Float64(Float64(fma(-0.13671875, Float64(Float64(x_m * x_m) / t_0), Float64(0.15625 / t_0)) * Float64(x_m * x_m)) - Float64(0.1875 / t_0)), Float64(x_m * x_m), Float64(0.25 / t_0)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_1) / Float64(1.0 + sqrt(t_1)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05], N[(N[(N[(N[(N[(-0.13671875 * N[(N[(x$95$m * x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(0.15625 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(0.1875 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.25 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 68.2%

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

      1. lift--.f64 N/A

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

      2. 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 68.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 67.2%

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

   if 1.05000000000000004 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. metadata-eval 98.5

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.13671875, x\_m \cdot x\_m, 0.15625\right) \cdot \left(x\_m \cdot x\_m\right) - 0.1875, x\_m \cdot x\_m, 0.25\right) \cdot \left(x\_m \cdot x\_m\right)}{1 + \sqrt{\left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)))
   (if (<= x_m 1.05)
     (/
      (*
       (fma
        (- (* (fma -0.13671875 (* x_m x_m) 0.15625) (* x_m x_m)) 0.1875)
        (* x_m x_m)
        0.25)
       (* x_m x_m))
      (+ 1.0 (sqrt (* (+ (cos (atan x_m)) 1.0) 0.5))))
     (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.05) {
		tmp = (fma(((fma(-0.13671875, (x_m * x_m), 0.15625) * (x_m * x_m)) - 0.1875), (x_m * x_m), 0.25) * (x_m * x_m)) / (1.0 + sqrt(((cos(atan(x_m)) + 1.0) * 0.5)));
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-0.13671875, Float64(x_m * x_m), 0.15625) * Float64(x_m * x_m)) - 0.1875), Float64(x_m * x_m), 0.25) * Float64(x_m * x_m)) / Float64(1.0 + sqrt(Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5))));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05], N[(N[(N[(N[(N[(N[(-0.13671875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.15625), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 68.2%

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

      1. lift--.f64 N/A

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

      2. 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 68.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   9. Applied rewrites 67.2%

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

   if 1.05000000000000004 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. metadata-eval 98.5

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.15625 \cdot \left(x\_m \cdot x\_m\right) - 0.1875, x\_m \cdot x\_m, 0.25\right) \cdot \left(x\_m \cdot x\_m\right)}{1 + \sqrt{\left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)))
   (if (<= x_m 1.15)
     (/
      (* (fma (- (* 0.15625 (* x_m x_m)) 0.1875) (* x_m x_m) 0.25) (* x_m x_m))
      (+ 1.0 (sqrt (* (+ (cos (atan x_m)) 1.0) 0.5))))
     (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.15) {
		tmp = (fma(((0.15625 * (x_m * x_m)) - 0.1875), (x_m * x_m), 0.25) * (x_m * x_m)) / (1.0 + sqrt(((cos(atan(x_m)) + 1.0) * 0.5)));
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(Float64(fma(Float64(Float64(0.15625 * Float64(x_m * x_m)) - 0.1875), Float64(x_m * x_m), 0.25) * Float64(x_m * x_m)) / Float64(1.0 + sqrt(Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5))));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.15], N[(N[(N[(N[(N[(0.15625 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 68.2%

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

      1. lift--.f64 N/A

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

      2. 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 68.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{32} \cdot {x}^{2} - \frac{3}{16}, {x}^{2}, \frac{1}{4}\right) \cdot {x}^{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{32} \cdot {x}^{2} - \frac{3}{16}, {x}^{2}, \frac{1}{4}\right) \cdot {x}^{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{32} \cdot \left(x \cdot x\right) - \frac{3}{16}, {x}^{2}, \frac{1}{4}\right) \cdot {x}^{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{32} \cdot \left(x \cdot x\right) - \frac{3}{16}, {x}^{2}, \frac{1}{4}\right) \cdot {x}^{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      10. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{32} \cdot \left(x \cdot x\right) - \frac{3}{16}, x \cdot x, \frac{1}{4}\right) \cdot {x}^{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{5}{32} \cdot \left(x \cdot x\right) - \frac{3}{16}, x \cdot x, \frac{1}{4}\right) \cdot {x}^{2}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot \frac{1}{2}}} \]

      12. pow2 N/A

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

      13. lift-*.f64 68.1

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

   9. Applied rewrites 68.1%

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

   if 1.1499999999999999 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. metadata-eval 98.5

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0025], N[(N[(N[(-0.1875 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $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], 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[Power[N[(x$95$m * x$95$m + 1.0), $MachinePrecision], -0.5], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00250000000000000005

   1. Initial program 68.1%

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

      1. lift--.f64 N/A

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

      2. 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 68.6%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 68.7

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

   6. Applied rewrites 68.7%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 66.9

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

   9. Applied rewrites 66.9%

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

   if 0.00250000000000000005 < x

   1. Initial program 97.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 99.3%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 99.3

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

   6. Applied rewrites 99.3%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. inv-pow N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. sqrt-pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-fma.f64 99.3

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

   8. Applied rewrites 99.3%

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

Alternative 6: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.08e-4

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if 1.08e-4 < x

   1. Initial program 97.9%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 99.3%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   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 99.3

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

   6. Applied rewrites 99.3%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. inv-pow N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. sqrt-pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-fma.f64 99.3

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

   8. Applied rewrites 99.3%

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

Alternative 7: 99.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)))
   (if (<= x_m 1.2) (* 0.125 (* x_m x_m)) (/ (- 1.0 t_0) (+ 1.0 (sqrt t_0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.2) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / x_m) + 0.5d0
    if (x_m <= 1.2d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = (1.0d0 - t_0) / (1.0d0 + sqrt(t_0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.2) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = (1.0 - t_0) / (1.0 + Math.sqrt(t_0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (0.5 / x_m) + 0.5
	tmp = 0
	if x_m <= 1.2:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = (1.0 - t_0) / (1.0 + math.sqrt(t_0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (0.5 / x_m) + 0.5;
	tmp = 0.0;
	if (x_m <= 1.2)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = (1.0 - t_0) / (1.0 + sqrt(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.2], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 68.2%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 1.19999999999999996 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. metadata-eval 98.5

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 8: 99.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118)
   (* 0.125 (* x_m x_m))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000118], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if 1.18e-4 < x

   1. Initial program 97.9%

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

      1. lift-hypot.f64 N/A

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

      2. metadata-eval N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 97.9

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

   4. Applied rewrites 97.9%

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

Alternative 9: 98.5% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 68.2%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 1.5 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. metadata-eval 98.5

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 98.9

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

   10. Applied rewrites 98.9%

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

Alternative 10: 97.7% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 68.2%

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

   3. Taylor expanded in x around 0

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

      1. sqrt-unprod N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. sqrt-undiv N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 36.8

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

   5. Applied rewrites 36.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 1.5 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.5%

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

Alternative 11: 51.8% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. lower--.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. sqrt-undiv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. pow2 N/A

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

   16. lower-*.f64 28.1

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

5. Applied rewrites 28.1%

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

6. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 50.7

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

8. Applied rewrites 50.7%

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

Alternative 12: 27.1% 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 76.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
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 26.9

      \[\leadsto 0 \]

5. Applied rewrites 26.9%

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

Reproduce

herbie shell --seed 2025072 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))