Given's Rotation SVD example, simplified

Percentage Accurate: 75.8% → 99.9%

Time: 4.7s

Alternatives: 16

Speedup: 1.0×

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: 75.8% 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, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5\\ \mathbf{if}\;x\_m \leq 0.027:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.25 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.15625 + -0.13671875 \cdot {x\_m}^{2}\right) - 0.1875\right)\right)}{1 + \sqrt{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{1 + \sqrt{\left(\sin \left(\tan^{-1} x\_m + 0.5 \cdot \pi\right) + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (Math.cos(Math.atan(x_m)) + 1.0) * 0.5;
	double tmp;
	if (x_m <= 0.027) {
		tmp = (Math.pow(x_m, 2.0) * (0.25 + (Math.pow(x_m, 2.0) * ((Math.pow(x_m, 2.0) * (0.15625 + (-0.13671875 * Math.pow(x_m, 2.0)))) - 0.1875)))) / (1.0 + Math.sqrt(t_0));
	} else {
		tmp = (1.0 - t_0) / (1.0 + Math.sqrt(((Math.sin((Math.atan(x_m) + (0.5 * Math.PI))) + 1.0) * 0.5)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (math.cos(math.atan(x_m)) + 1.0) * 0.5
	tmp = 0
	if x_m <= 0.027:
		tmp = (math.pow(x_m, 2.0) * (0.25 + (math.pow(x_m, 2.0) * ((math.pow(x_m, 2.0) * (0.15625 + (-0.13671875 * math.pow(x_m, 2.0)))) - 0.1875)))) / (1.0 + math.sqrt(t_0))
	else:
		tmp = (1.0 - t_0) / (1.0 + math.sqrt(((math.sin((math.atan(x_m) + (0.5 * math.pi))) + 1.0) * 0.5)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)
	tmp = 0.0
	if (x_m <= 0.027)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.25 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.15625 + Float64(-0.13671875 * (x_m ^ 2.0)))) - 0.1875)))) / Float64(1.0 + sqrt(t_0)));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(1.0 + sqrt(Float64(Float64(sin(Float64(atan(x_m) + Float64(0.5 * pi))) + 1.0) * 0.5))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (cos(atan(x_m)) + 1.0) * 0.5;
	tmp = 0.0;
	if (x_m <= 0.027)
		tmp = ((x_m ^ 2.0) * (0.25 + ((x_m ^ 2.0) * (((x_m ^ 2.0) * (0.15625 + (-0.13671875 * (x_m ^ 2.0)))) - 0.1875)))) / (1.0 + sqrt(t_0));
	else
		tmp = (1.0 - t_0) / (1.0 + sqrt(((sin((atan(x_m) + (0.5 * pi))) + 1.0) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.0269999999999999997

   1. Initial program 53.6%

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

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

   3. Applied rewrites 53.6%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   4. 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. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 100.0

          \[\leadsto \frac{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.15625 + -0.13671875 \cdot {x}^{2}\right) - 0.1875\right)\right)}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.15625 + -0.13671875 \cdot {x}^{2}\right) - 0.1875\right)\right)}}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}} \]

   if 0.0269999999999999997 < x

   1. Initial program 98.4%

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

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

   3. Applied rewrites 99.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0136:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.25 + {x\_m}^{2} \cdot \left({x\_m}^{2} \cdot \left(0.15625 + -0.13671875 \cdot {x\_m}^{2}\right) - 0.1875\right)\right)}{1 + \sqrt{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.1875 - 0.25, x\_m \cdot x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}{1 + \sqrt{\left(\sin \left(\tan^{-1} x\_m + 0.5 \cdot \pi\right) + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0136)
   (/
    (*
     (pow x_m 2.0)
     (+
      0.25
      (*
       (pow x_m 2.0)
       (-
        (* (pow x_m 2.0) (+ 0.15625 (* -0.13671875 (pow x_m 2.0))))
        0.1875))))
    (+ 1.0 (sqrt (fma (- (* (* x_m x_m) 0.1875) 0.25) (* x_m x_m) 1.0))))
   (/
    (- 1.0 (* (+ (cos (atan x_m)) 1.0) 0.5))
    (+ 1.0 (sqrt (* (+ (sin (+ (atan x_m) (* 0.5 PI))) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0136) {
		tmp = (pow(x_m, 2.0) * (0.25 + (pow(x_m, 2.0) * ((pow(x_m, 2.0) * (0.15625 + (-0.13671875 * pow(x_m, 2.0)))) - 0.1875)))) / (1.0 + sqrt(fma((((x_m * x_m) * 0.1875) - 0.25), (x_m * x_m), 1.0)));
	} else {
		tmp = (1.0 - ((cos(atan(x_m)) + 1.0) * 0.5)) / (1.0 + sqrt(((sin((atan(x_m) + (0.5 * ((double) M_PI)))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0136)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.25 + Float64((x_m ^ 2.0) * Float64(Float64((x_m ^ 2.0) * Float64(0.15625 + Float64(-0.13671875 * (x_m ^ 2.0)))) - 0.1875)))) / Float64(1.0 + sqrt(fma(Float64(Float64(Float64(x_m * x_m) * 0.1875) - 0.25), Float64(x_m * x_m), 1.0))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(sin(Float64(atan(x_m) + Float64(0.5 * pi))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0136], N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.15625 + N[(-0.13671875 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1875), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Sin[N[(N[ArcTan[x$95$m], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0135999999999999992

   1. Initial program 53.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.5

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 53.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 99.9

          \[\leadsto \frac{{x}^{2} \cdot \left(0.25 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.15625 + -0.13671875 \cdot {x}^{2}\right) - 0.1875\right)\right)}{1 + \sqrt{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.1875 - 0.25, x \cdot x, 1\right)}} \]

   9. Applied rewrites 99.9%

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

   if 0.0135999999999999992 < x

   1. Initial program 98.4%

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

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

   3. Applied rewrites 99.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0115:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.25 + {x\_m}^{2} \cdot \left(0.15625 \cdot {x\_m}^{2} - 0.1875\right)\right)}{1 + \sqrt{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.1875 - 0.25, x\_m \cdot x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}{1 + \sqrt{\left(\sin \left(\tan^{-1} x\_m + 0.5 \cdot \pi\right) + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0115)
   (/
    (*
     (pow x_m 2.0)
     (+ 0.25 (* (pow x_m 2.0) (- (* 0.15625 (pow x_m 2.0)) 0.1875))))
    (+ 1.0 (sqrt (fma (- (* (* x_m x_m) 0.1875) 0.25) (* x_m x_m) 1.0))))
   (/
    (- 1.0 (* (+ (cos (atan x_m)) 1.0) 0.5))
    (+ 1.0 (sqrt (* (+ (sin (+ (atan x_m) (* 0.5 PI))) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0115) {
		tmp = (pow(x_m, 2.0) * (0.25 + (pow(x_m, 2.0) * ((0.15625 * pow(x_m, 2.0)) - 0.1875)))) / (1.0 + sqrt(fma((((x_m * x_m) * 0.1875) - 0.25), (x_m * x_m), 1.0)));
	} else {
		tmp = (1.0 - ((cos(atan(x_m)) + 1.0) * 0.5)) / (1.0 + sqrt(((sin((atan(x_m) + (0.5 * ((double) M_PI)))) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0115)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.25 + Float64((x_m ^ 2.0) * Float64(Float64(0.15625 * (x_m ^ 2.0)) - 0.1875)))) / Float64(1.0 + sqrt(fma(Float64(Float64(Float64(x_m * x_m) * 0.1875) - 0.25), Float64(x_m * x_m), 1.0))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(cos(atan(x_m)) + 1.0) * 0.5)) / Float64(1.0 + sqrt(Float64(Float64(sin(Float64(atan(x_m) + Float64(0.5 * pi))) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0115], N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.15625 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1875), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[Sin[N[(N[ArcTan[x$95$m], $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0115

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.5

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 53.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 0.0115 < x

   1. Initial program 98.4%

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

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

   3. Applied rewrites 99.9%

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

      1. lift-atan.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-PI.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 99.9

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

   8. Applied rewrites 99.9%

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

Alternative 4: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0115:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.25 + {x\_m}^{2} \cdot \left(0.15625 \cdot {x\_m}^{2} - 0.1875\right)\right)}{1 + \sqrt{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.1875 - 0.25, x\_m \cdot x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0115)
   (/
    (*
     (pow x_m 2.0)
     (+ 0.25 (* (pow x_m 2.0) (- (* 0.15625 (pow x_m 2.0)) 0.1875))))
    (+ 1.0 (sqrt (fma (- (* (* x_m x_m) 0.1875) 0.25) (* x_m x_m) 1.0))))
   (/
    (- 1.0 (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5))
    (+ 1.0 (sqrt (* (+ (cos (atan x_m)) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0115) {
		tmp = (pow(x_m, 2.0) * (0.25 + (pow(x_m, 2.0) * ((0.15625 * pow(x_m, 2.0)) - 0.1875)))) / (1.0 + sqrt(fma((((x_m * x_m) * 0.1875) - 0.25), (x_m * x_m), 1.0)));
	} else {
		tmp = (1.0 - ((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / (1.0 + sqrt(((cos(atan(x_m)) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0115)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.25 + Float64((x_m ^ 2.0) * Float64(Float64(0.15625 * (x_m ^ 2.0)) - 0.1875)))) / Float64(1.0 + sqrt(fma(Float64(Float64(Float64(x_m * x_m) * 0.1875) - 0.25), Float64(x_m * x_m), 1.0))));
	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(cos(atan(x_m)) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0115], N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.25 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(N[(0.15625 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.1875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1875), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $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[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0115

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.5

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 53.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 0.0115 < x

   1. Initial program 98.4%

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

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

   3. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   4. 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. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0022:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.25 + -0.1875 \cdot {x\_m}^{2}\right)}{1 + \sqrt{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.1875 - 0.25, x\_m \cdot x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(\sqrt{\frac{1}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}} + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x\_m + 1\right) \cdot 0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0022)
   (/
    (* (pow x_m 2.0) (+ 0.25 (* -0.1875 (pow x_m 2.0))))
    (+ 1.0 (sqrt (fma (- (* (* x_m x_m) 0.1875) 0.25) (* x_m x_m) 1.0))))
   (/
    (- 1.0 (* (+ (sqrt (/ 1.0 (fma x_m x_m 1.0))) 1.0) 0.5))
    (+ 1.0 (sqrt (* (+ (cos (atan x_m)) 1.0) 0.5))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0022) {
		tmp = (pow(x_m, 2.0) * (0.25 + (-0.1875 * pow(x_m, 2.0)))) / (1.0 + sqrt(fma((((x_m * x_m) * 0.1875) - 0.25), (x_m * x_m), 1.0)));
	} else {
		tmp = (1.0 - ((sqrt((1.0 / fma(x_m, x_m, 1.0))) + 1.0) * 0.5)) / (1.0 + sqrt(((cos(atan(x_m)) + 1.0) * 0.5)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0022)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.25 + Float64(-0.1875 * (x_m ^ 2.0)))) / Float64(1.0 + sqrt(fma(Float64(Float64(Float64(x_m * x_m) * 0.1875) - 0.25), Float64(x_m * x_m), 1.0))));
	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(cos(atan(x_m)) + 1.0) * 0.5))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0022], N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.25 + N[(-0.1875 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1875), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $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[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00220000000000000013

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.5

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 53.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 0.00220000000000000013 < x

   1. Initial program 98.3%

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

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

   3. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1 - \left(\cos \tan^{-1} x + 1\right) \cdot 0.5}{1 + \sqrt{\left(\cos \tan^{-1} x + 1\right) \cdot 0.5}}} \]
   4. 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. sqrt-undiv N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 6: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0024:\\ \;\;\;\;\frac{{x\_m}^{2} \cdot \left(0.25 + -0.1875 \cdot {x\_m}^{2}\right)}{1 + \sqrt{\mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.1875 - 0.25, x\_m \cdot x\_m, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0024)
   (/
    (* (pow x_m 2.0) (+ 0.25 (* -0.1875 (pow x_m 2.0))))
    (+ 1.0 (sqrt (fma (- (* (* x_m x_m) 0.1875) 0.25) (* x_m x_m) 1.0))))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0024) {
		tmp = (pow(x_m, 2.0) * (0.25 + (-0.1875 * pow(x_m, 2.0)))) / (1.0 + sqrt(fma((((x_m * x_m) * 0.1875) - 0.25), (x_m * x_m), 1.0)));
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0024)
		tmp = Float64(Float64((x_m ^ 2.0) * Float64(0.25 + Float64(-0.1875 * (x_m ^ 2.0)))) / Float64(1.0 + sqrt(fma(Float64(Float64(Float64(x_m * x_m) * 0.1875) - 0.25), Float64(x_m * x_m), 1.0))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0024], N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.25 + N[(-0.1875 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.1875), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00239999999999999979

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.5

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.5%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 53.5%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if 0.00239999999999999979 < x

   1. Initial program 98.3%

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

Alternative 7: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.00014:\\ \;\;\;\;0.25 \cdot \frac{{x\_m}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00014)
   (* 0.25 (/ (pow x_m 2.0) (+ 1.0 (* (sqrt 0.5) (sqrt 2.0)))))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00014) {
		tmp = 0.25 * (pow(x_m, 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.00014) {
		tmp = 0.25 * (Math.pow(x_m, 2.0) / (1.0 + (Math.sqrt(0.5) * Math.sqrt(2.0))));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 * (1.0 + (1.0 / Math.hypot(1.0, x_m)))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.00014:
		tmp = 0.25 * (math.pow(x_m, 2.0) / (1.0 + (math.sqrt(0.5) * math.sqrt(2.0))))
	else:
		tmp = 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x_m)))))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00014)
		tmp = Float64(0.25 * Float64((x_m ^ 2.0) / Float64(1.0 + Float64(sqrt(0.5) * sqrt(2.0)))));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m))))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.00014)
		tmp = 0.25 * ((x_m ^ 2.0) / (1.0 + (sqrt(0.5) * sqrt(2.0))));
	else
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00014], N[(0.25 * N[(N[Power[x$95$m, 2.0], $MachinePrecision] / N[(1.0 + N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999e-4

   1. Initial program 53.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.4

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.4%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 99.7

         \[\leadsto 0.25 \cdot \frac{{x}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}} \]

   9. Applied rewrites 99.7%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{1 + \sqrt{0.5} \cdot \sqrt{2}}} \]

   if 1.3999999999999999e-4 < x

   1. Initial program 98.1%

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

Alternative 8: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

Alternative 9: 75.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.8%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. 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 75.8

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

3. Applied rewrites 75.8%

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

Alternative 10: 75.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x\_m \cdot x\_m\right) - 0.25, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (- 1.0 (sqrt (fma (- (* 0.1875 (* x_m x_m)) 0.25) (* x_m x_m) 1.0)))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = 1.0 - sqrt(fma(((0.1875 * (x_m * x_m)) - 0.25), (x_m * x_m), 1.0));
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(1.0 - sqrt(fma(Float64(Float64(0.1875 * Float64(x_m * x_m)) - 0.25), Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(1.0 - N[Sqrt[N[(N[(N[(0.1875 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 53.4

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 53.4%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

Alternative 11: 75.1% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(1.0 - sqrt(fma(-0.25, Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.2], N[(1.0 - N[Sqrt[N[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 53.2

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

   4. Applied rewrites 53.2%

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

   if 1.19999999999999996 < x

   1. Initial program 98.5%

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

   2. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.7

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

   4. Applied rewrites 97.7%

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

Alternative 12: 75.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.42:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(-0.25, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.42)
   (- 1.0 (sqrt (fma -0.25 (* x_m x_m) 1.0)))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.42) {
		tmp = 1.0 - sqrt(fma(-0.25, (x_m * x_m), 1.0));
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.42)
		tmp = Float64(1.0 - sqrt(fma(-0.25, Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.42], N[(1.0 - N[Sqrt[N[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $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 < 1.4199999999999999

   1. Initial program 53.8%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 53.1

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

   4. Applied rewrites 53.1%

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

   if 1.4199999999999999 < x

   1. Initial program 98.5%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 0.8

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 0.3%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 97.8

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

   9. Applied rewrites 97.8%

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

Alternative 13: 75.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;1 - \mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.5, -0.25, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.5)
   (- 1.0 (fma (* (* x_m x_m) 0.5) -0.25 1.0))
   (/ 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 = 1.0 - fma(((x_m * x_m) * 0.5), -0.25, 1.0);
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(1.0 - fma(Float64(Float64(x_m * x_m) * 0.5), -0.25, 1.0));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.5], N[(1.0 - N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * -0.25 + 1.0), $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 53.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sqrt-undiv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 53.1

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

   4. Applied rewrites 53.1%

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

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 0.8

         \[\leadsto 1 - \sqrt{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot \color{blue}{x}, 1\right)} \]

   4. Applied rewrites 0.8%

      \[\leadsto 1 - \sqrt{\color{blue}{\mathsf{fma}\left(0.1875 \cdot \left(x \cdot x\right) - 0.25, x \cdot x, 1\right)}} \]

   6. Applied rewrites 0.3%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 97.8

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

   9. Applied rewrites 97.8%

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

Alternative 14: 74.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5:\\ \;\;\;\;1 - \mathsf{fma}\left(\left(x\_m \cdot x\_m\right) \cdot 0.5, -0.25, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.5)
   (- 1.0 (fma (* (* x_m x_m) 0.5) -0.25 1.0))
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.5) {
		tmp = 1.0 - fma(((x_m * x_m) * 0.5), -0.25, 1.0);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.5)
		tmp = Float64(1.0 - fma(Float64(Float64(x_m * x_m) * 0.5), -0.25, 1.0));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.5], N[(1.0 - N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * -0.25 + 1.0), $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 53.8%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sqrt-undiv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. sqrt-unprod N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lower-*.f64 53.1

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

   4. Applied rewrites 53.1%

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

   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. Taylor expanded in x around inf

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

   4. Applied rewrites 96.3%

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

Alternative 15: 74.3% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 - \sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   3. 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 67.3

         \[\leadsto 0 \]

   4. Applied rewrites 67.3%

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

   if 2.1500000000000001e-77 < x

   1. Initial program 81.2%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 78.7%

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

Alternative 16: 28.0% 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 75.8%

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

2. Taylor expanded in x around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

      \[\leadsto 1 - 1 \]

   4. metadata-eval 28.0

      \[\leadsto 0 \]

4. Applied rewrites 28.0%

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

Reproduce

herbie shell --seed 2025105 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))