Given's Rotation SVD example, simplified

Percentage Accurate: 76.2% → 99.8%

Time: 9.3s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}, -0.5, 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x, 0.5, 0.5\right)} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.002)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (/
    (fma (pow (fma x x 1.0) -0.5) -0.5 0.5)
    (+ (sqrt (fma (cos (atan x)) 0.5 0.5)) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = fma(pow(fma(x, x, 1.0), -0.5), -0.5, 0.5) / (sqrt(fma(cos(atan(x)), 0.5, 0.5)) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.002)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(fma((fma(x, x, 1.0) ^ -0.5), -0.5, 0.5) / Float64(sqrt(fma(cos(atan(x)), 0.5, 0.5)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.002], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Power[N[(x * x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[(N[Sqrt[N[(N[Cos[N[ArcTan[x], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.002

   1. Initial program 53.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 100.0

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 98.4

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

   9. Applied rewrites 98.4%

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

   11. Applied rewrites 99.9%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1 - \frac{\frac{0.5}{x}}{x}}{x}, 0.5, 0.5\right)\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_0}^{1.5}}{\left(1 + t\_0\right) + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (/ (- 1.0 (/ (/ 0.5 x) x)) x) 0.5 0.5)))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (fma
       (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/ (- 1.0 (pow t_0 1.5)) (+ (+ 1.0 t_0) (sqrt t_0))))))

C

double code(double x) {
	double t_0 = fma(((1.0 - ((0.5 / x) / x)) / x), 0.5, 0.5);
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma((((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (1.0 - pow(t_0, 1.5)) / ((1.0 + t_0) + sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(Float64(1.0 - Float64(Float64(0.5 / x) / x)) / x), 0.5, 0.5)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - (t_0 ^ 1.5)) / Float64(Float64(1.0 + t_0) + sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$0, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$0), $MachinePrecision] + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 98.5%

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

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1 - \frac{\frac{0.5}{x}}{x}}{x}, 0.5, 0.5\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0 \cdot t\_1}{\left(1 + t\_0\right) + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (/ (- 1.0 (/ (/ 0.5 x) x)) x) 0.5 0.5)) (t_1 (sqrt t_0)))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (fma
       (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/ (- 1.0 (* t_0 t_1)) (+ (+ 1.0 t_0) t_1)))))

C

double code(double x) {
	double t_0 = fma(((1.0 - ((0.5 / x) / x)) / x), 0.5, 0.5);
	double t_1 = sqrt(t_0);
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma((((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (1.0 - (t_0 * t_1)) / ((1.0 + t_0) + t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(Float64(1.0 - Float64(Float64(0.5 / x) / x)) / x), 0.5, 0.5)
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - Float64(t_0 * t_1)) / Float64(Float64(1.0 + t_0) + t_1));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(1.0 - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 98.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. unpow3 N/A

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

   9. Applied rewrites 98.5%

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

Alternative 4: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.002:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + {\left(\mathsf{fma}\left(x, x, 1\right)\right)}^{-0.5}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.002)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (pow (fma x x 1.0) -0.5)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.002) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + pow(fma(x, x, 1.0), -0.5))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.002)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + (fma(x, x, 1.0) ^ -0.5)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.002], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[Power[N[(x * x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.002

   1. Initial program 53.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 100.0

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 1.002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

      6. pow-prod-down N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

Alternative 5: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 - \frac{\frac{0.5}{x}}{x}}{x}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\sqrt{0.5} \cdot \sqrt{\left(t\_0 + 1\right) \cdot \mathsf{fma}\left(t\_0, 0.5, 0.5\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- 1.0 (/ (/ 0.5 x) x)) x)))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (fma
       (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
       (* x x)
       0.125)
      (* x x))
     (- 1.0 (sqrt (* (sqrt 0.5) (sqrt (* (+ t_0 1.0) (fma t_0 0.5 0.5)))))))))

C

double code(double x) {
	double t_0 = (1.0 - ((0.5 / x) / x)) / x;
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma((((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = 1.0 - sqrt((sqrt(0.5) * sqrt(((t_0 + 1.0) * fma(t_0, 0.5, 0.5)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(1.0 - Float64(Float64(0.5 / x) / x)) / x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(1.0 - sqrt(Float64(sqrt(0.5) * sqrt(Float64(Float64(t_0 + 1.0) * fma(t_0, 0.5, 0.5))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(1.0 - N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[Sqrt[0.5], $MachinePrecision] * N[Sqrt[N[(N[(t$95$0 + 1.0), $MachinePrecision] * N[(t$95$0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

      1. rem-square-sqrt N/A

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

      2. lift-*.f64 N/A

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

      5. lift-sqrt.f64 N/A

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

   7. Applied rewrites 97.0%

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

Alternative 6: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1 - \frac{0.5}{x \cdot x}}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (fma
     (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
     (* x x)
     0.125)
    (* x x))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ (- 1.0 (/ 0.5 (* x x))) x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma((((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + ((1.0 - (0.5 / (x * x))) / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(Float64(1.0 - Float64(0.5 / Float64(x * x))) / x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(N[(1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 97.0%

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

Alternative 7: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (fma
     (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
     (* x x)
     0.125)
    (* x x))
   (/ (- 1.0 0.5) (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma((((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (1.0 - 0.5) / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - 0.5) / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - 0.5), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 97.0%

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

Alternative 8: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma (- (* 0.0673828125 (* x x)) 0.0859375) (* x x) 0.125) x) x)
   (/ (- 1.0 0.5) (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(((0.0673828125 * (x * x)) - 0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = (1.0 - 0.5) / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x * x)) - 0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(1.0 - 0.5) / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(0.0673828125 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 - 0.5), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right) \cdot {x}^{2}} + \frac{1}{8}\right) \cdot x\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      9. lower--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{69}{1024} \cdot {x}^{2}} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      11. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{11}{128}, {x}^{2}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      13. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{69}{1024} \cdot \left(x \cdot x\right) - \frac{11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      14. lower-*.f64 99.6

          \[\leadsto \left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   10. Applied rewrites 99.6%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 97.0%

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

Alternative 9: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (/ (- 1.0 0.5) (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = (1.0 - 0.5) / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(1.0 - 0.5) / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(1.0 - 0.5), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.5

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 97.0%

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

Alternative 10: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (- 1.0 (sqrt (+ (/ 0.5 x) 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt(((0.5 / x) + 0.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x) + 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.5

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 96.6

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

   5. Applied rewrites 96.6%

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

Alternative 11: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)
   (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-11}{128} \cdot {x}^{2} + \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-11}{128}, {x}^{2}, \frac{1}{8}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-11}{128}, \color{blue}{x \cdot x}, \frac{1}{8}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.5

         \[\leadsto \left(\mathsf{fma}\left(-0.0859375, \color{blue}{x \cdot x}, 0.125\right) \cdot x\right) \cdot x \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

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

Alternative 12: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* 0.125 (* x x)) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = 0.125 * (x * x)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = 0.125 * (x * x);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 54.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 0.8

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

   5. Applied rewrites 0.8%

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.1

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

   10. Applied rewrites 99.1%

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

   if 2 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.5%

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

Alternative 13: 51.5% accurate, 12.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.125 (* x x)))

C

double code(double x) {
	return 0.125 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.125d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.125 * (x * x);
}

Python

def code(x):
	return 0.125 * (x * x)

Julia

function code(x)
	return Float64(0.125 * Float64(x * x))
end

MATLAB

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

Wolfram

code[x_] := N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 46.6

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

5. Applied rewrites 46.6%

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

7. Applied rewrites 47.1%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 53.9

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

10. Applied rewrites 53.9%

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

Reproduce

herbie shell --seed 2024326 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))