Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.9%

Time: 7.7s

Alternatives: 16

Speedup: 6.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.25}{\mathsf{fma}\left(x\_m, x\_m, 1\right)} + 0.25\\ t_1 := \frac{-0.5}{\mathsf{hypot}\left(1, x\_m\right)}\\ t_2 := 0.5 - t\_1\\ \mathbf{if}\;x\_m \leq 0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.0625}{t\_0} - \frac{{t\_1}^{4}}{t\_0}}{t\_2}}{\sqrt{t\_2} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.25 (fma x_m x_m 1.0)) 0.25))
        (t_1 (/ -0.5 (hypot 1.0 x_m)))
        (t_2 (- 0.5 t_1)))
   (if (<= x_m 0.01)
     (*
      (*
       (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
       x_m)
      x_m)
     (/
      (/ (- (/ 0.0625 t_0) (/ (pow t_1 4.0) t_0)) t_2)
      (- (sqrt t_2) -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.25 / fma(x_m, x_m, 1.0)) + 0.25;
	double t_1 = -0.5 / hypot(1.0, x_m);
	double t_2 = 0.5 - t_1;
	double tmp;
	if (x_m <= 0.01) {
		tmp = (fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (((0.0625 / t_0) - (pow(t_1, 4.0) / t_0)) / t_2) / (sqrt(t_2) - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.25 / fma(x_m, x_m, 1.0)) + 0.25)
	t_1 = Float64(-0.5 / hypot(1.0, x_m))
	t_2 = Float64(0.5 - t_1)
	tmp = 0.0
	if (x_m <= 0.01)
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(Float64(Float64(0.0625 / t_0) - Float64((t_1 ^ 4.0) / t_0)) / t_2) / Float64(sqrt(t_2) - -1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.25 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + 0.25), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - t$95$1), $MachinePrecision]}, If[LessEqual[x$95$m, 0.01], N[(N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(0.0625 / t$95$0), $MachinePrecision] - N[(N[Power[t$95$1, 4.0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[Sqrt[t$95$2], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0100000000000000002

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. 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 63.1

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

   7. Applied rewrites 63.1%

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

   if 0.0100000000000000002 < x

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

   10. Applied rewrites 99.9%

       \[\leadsto \frac{\frac{\color{blue}{\frac{0.0625}{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25} - \frac{{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4}}{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25}}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.0625}{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25} - \frac{{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{4}}{\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\_m\right)}\\ \mathbf{if}\;x\_m \leq 0.0095:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}{t\_0}}{\sqrt{t\_0} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x_m)))))
   (if (<= x_m 0.0095)
     (*
      (*
       (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
       x_m)
      x_m)
     (/ (/ (- 0.25 (/ 0.25 (fma x_m x_m 1.0))) t_0) (- (sqrt t_0) -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x_m));
	double tmp;
	if (x_m <= 0.0095) {
		tmp = (fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = ((0.25 - (0.25 / fma(x_m, x_m, 1.0))) / t_0) / (sqrt(t_0) - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x_m)))
	tmp = 0.0
	if (x_m <= 0.0095)
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(Float64(0.25 - Float64(0.25 / fma(x_m, x_m, 1.0))) / t_0) / Float64(sqrt(t_0) - -1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0095], N[(N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(0.25 - N[(0.25 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00949999999999999976

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. 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 63.1

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

   7. Applied rewrites 63.1%

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

   if 0.00949999999999999976 < x

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0095:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\_m\right)}\\ \mathbf{if}\;x\_m \leq 0.0095:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}{t\_0 \cdot \left(\sqrt{t\_0} - -1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x_m)))))
   (if (<= x_m 0.0095)
     (*
      (*
       (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
       x_m)
      x_m)
     (/ (- 0.25 (/ 0.25 (fma x_m x_m 1.0))) (* t_0 (- (sqrt t_0) -1.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x_m));
	double tmp;
	if (x_m <= 0.0095) {
		tmp = (fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (0.25 - (0.25 / fma(x_m, x_m, 1.0))) / (t_0 * (sqrt(t_0) - -1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x_m)))
	tmp = 0.0
	if (x_m <= 0.0095)
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(0.25 - Float64(0.25 / fma(x_m, x_m, 1.0))) / Float64(t_0 * Float64(sqrt(t_0) - -1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0095], N[(N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(0.25 - N[(0.25 / N[(x$95$m * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[Sqrt[t$95$0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00949999999999999976

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. 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 63.1

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

   7. Applied rewrites 63.1%

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

   if 0.00949999999999999976 < x

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

   10. Applied rewrites 99.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0095:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + {\left(\mathsf{hypot}\left(1, x\_m\right)\right)}^{-1}\right)} \leq 0.8:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (pow (hypot 1.0 x_m) -1.0)))) 0.8)
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))
   (*
    (fma
     (-
      (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
      0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + pow(hypot(1.0, x_m), -1.0)))) <= 0.8) {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	} else {
		tmp = fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + (hypot(1.0, x_m) ^ -1.0)))) <= 0.8)
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	else
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[Power[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   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 96.9

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

   5. Applied rewrites 96.9%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 55.1

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

   4. Applied rewrites 55.1%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 55.2

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

   6. Applied rewrites 55.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   10. 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

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

   11. Applied rewrites 99.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1}\right)} \leq 0.8:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\_m\right)}\\ \mathbf{if}\;x\_m \leq 0.011:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 0.5}{\sqrt{0.5 - t\_0} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x_m))))
   (if (<= x_m 0.011)
     (*
      (*
       (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
       x_m)
      x_m)
     (/ (+ t_0 0.5) (- (sqrt (- 0.5 t_0)) -1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = -0.5 / hypot(1.0, x_m);
	double tmp;
	if (x_m <= 0.011) {
		tmp = (fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (t_0 + 0.5) / (sqrt((0.5 - t_0)) - -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(-0.5 / hypot(1.0, x_m))
	tmp = 0.0
	if (x_m <= 0.011)
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(t_0 + 0.5) / Float64(sqrt(Float64(0.5 - t_0)) - -1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.011], N[(N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(t$95$0 + 0.5), $MachinePrecision] / N[(N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.010999999999999999

   1. Initial program 71.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. 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 63.1

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

   7. Applied rewrites 63.1%

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

   if 0.010999999999999999 < x

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.011:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + {\left(\mathsf{hypot}\left(1, x\_m\right)\right)}^{-1}\right)} \leq 0.8:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (sqrt (* 0.5 (+ 1.0 (pow (hypot 1.0 x_m) -1.0)))) 0.8)
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))
   (*
    (* (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125) x_m)
    x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (sqrt((0.5 * (1.0 + pow(hypot(1.0, x_m), -1.0)))) <= 0.8) {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	} else {
		tmp = (fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + (hypot(1.0, x_m) ^ -1.0)))) <= 0.8)
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	else
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[Power[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.8], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))) < 0.80000000000000004

   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 96.9

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

   5. Applied rewrites 96.9%

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

   if 0.80000000000000004 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))

   1. Initial program 55.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 55.1

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left(\frac{69}{1024} \cdot {x}^{2} - \frac{11}{128}\right)\right)} \]
   6. 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 99.1

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

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1}\right)} \leq 0.8:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 0.5 - \frac{\frac{0.25}{x\_m \cdot x\_m} - 0.5}{x\_m}\\ \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ (- (/ 0.25 (* x_m x_m)) 0.5) x_m))))
   (if (<= x_m 1.0)
     (*
      (fma
       (-
        (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
        0.0859375)
       (* x_m x_m)
       0.125)
      (* x_m x_m))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.5 - (((0.25 / (x_m * x_m)) - 0.5) / x_m);
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.5 - Float64(Float64(Float64(0.25 / Float64(x_m * x_m)) - 0.5) / x_m))
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.5 - N[(N[(N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 72.1

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

   6. Applied rewrites 72.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   10. 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

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

   11. Applied rewrites 62.0%

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

   if 1 < x

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

         \[\leadsto 1 - \sqrt{0.5 - \frac{\frac{0.25}{\color{blue}{x \cdot x}} - 0.5}{x}} \]

   7. Applied rewrites 98.6%

      \[\leadsto 1 - \sqrt{0.5 - \color{blue}{\frac{\frac{0.25}{x \cdot x} - 0.5}{x}}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 100.0%

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

Alternative 8: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt{0.5}}{x\_m}, -0.5, \frac{0.5}{\sqrt{0.5} + 1}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.15)
   (*
    (fma
     (-
      (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
      0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))
   (fma (/ (sqrt 0.5) x_m) -0.5 (/ 0.5 (+ (sqrt 0.5) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.15) {
		tmp = fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = fma((sqrt(0.5) / x_m), -0.5, (0.5 / (sqrt(0.5) + 1.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.15)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = fma(Float64(sqrt(0.5) / x_m), -0.5, Float64(0.5 / Float64(sqrt(0.5) + 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.15], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] / x$95$m), $MachinePrecision] * -0.5 + N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 72.1

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

   6. Applied rewrites 72.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   10. 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

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

   11. Applied rewrites 62.0%

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

   if 1.1499999999999999 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 97.2

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 98.7%

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

Alternative 9: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{\frac{0.25}{x\_m \cdot x\_m} - 0.5}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0)
   (*
    (fma
     (-
      (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
      0.0859375)
     (* x_m x_m)
     0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (- 0.5 (/ (- (/ 0.25 (* x_m x_m)) 0.5) x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt((0.5 - (((0.25 / (x_m * x_m)) - 0.5) / x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 - Float64(Float64(Float64(0.25 / Float64(x_m * x_m)) - 0.5) / x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 - N[(N[(N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 72.1

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

   6. Applied rewrites 72.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--r- N/A

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

      4. metadata-eval N/A

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

      5. flip-+ N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{69}{1024} + \frac{-1843}{32768} \cdot {x}^{2}\right) - \frac{11}{128}\right)\right)} \]
   10. 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

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

   11. Applied rewrites 62.0%

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

   if 1 < x

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

         \[\leadsto 1 - \sqrt{0.5 - \frac{\frac{0.25}{\color{blue}{x \cdot x}} - 0.5}{x}} \]

   7. Applied rewrites 98.6%

      \[\leadsto 1 - \sqrt{0.5 - \color{blue}{\frac{\frac{0.25}{x \cdot x} - 0.5}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma (* x_m x_m) -0.0859375 0.125) (* x_m x_m))
   (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma((x_m * x_m), -0.0859375, 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. 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 61.7

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

   7. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.7%

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

   if 1.1000000000000001 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   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 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 11: 98.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma (* x_m x_m) -0.0859375 0.125) (* x_m x_m))
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma((x_m * x_m), -0.0859375, 0.125) * (x_m * x_m);
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. 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 61.7

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

   7. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.7%

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

   if 1.1000000000000001 < x

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.6

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

   4. Applied rewrites 98.6%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 96.1

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

   9. Applied rewrites 96.1%

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

Alternative 12: 98.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.0859375, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (fma (* x_m x_m) -0.0859375 0.125) (* x_m x_m))
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = fma((x_m * x_m), -0.0859375, 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(fma(Float64(x_m * x_m), -0.0859375, 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. 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 61.7

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

   7. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.7%

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

   if 1.1000000000000001 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.7%

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

Alternative 13: 98.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (* (* (fma -0.0859375 (* x_m x_m) 0.125) x_m) x_m)
   (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = (fma(-0.0859375, (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(fma(-0.0859375, Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(-0.0859375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   6. 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 61.7

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

   7. Applied rewrites 61.7%

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

   if 1.1000000000000001 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.7%

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

Alternative 14: 97.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* (* x_m x_m) 0.125) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 71.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 71.5

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

   4. Applied rewrites 71.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 62.7

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

   7. Applied rewrites 62.7%

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

   if 1.55000000000000004 < x

   1. Initial program 98.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.7%

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

Alternative 15: 51.1% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 78.2

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

4. Applied rewrites 78.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 48.4

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

7. Applied rewrites 48.4%

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

Alternative 16: 26.5% accurate, 33.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (- 1.0 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0 - 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 - 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 - 1.0

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - 1.0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 - 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

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

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 78.2

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

4. Applied rewrites 78.2%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 26.6%

   \[\leadsto 1 - \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))