Given's Rotation SVD example, simplified

Percentage Accurate: 76.3% → 99.9%

Time: 10.5s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 - t\_0}{0.125 + \frac{0.125}{\mathsf{fma}\left(x, x, 1\right) \cdot t\_1}} \cdot \frac{\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{t\_1}}}}{\frac{1}{\left(0.25 + t\_0\right) - \frac{0.25}{t\_1}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.25 (fma x x 1.0))) (t_1 (sqrt (fma x x 1.0))))
   (if (<= (hypot 1.0 x) 1.02)
     (*
      (* x x)
      (fma
       (* x x)
       (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
       0.125))
     (*
      (/ (- 0.25 t_0) (+ 0.125 (/ 0.125 (* (fma x x 1.0) t_1))))
      (/
       (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 t_1)))))
       (/ 1.0 (- (+ 0.25 t_0) (/ 0.25 t_1))))))))

C

double code(double x) {
	double t_0 = 0.25 / fma(x, x, 1.0);
	double t_1 = sqrt(fma(x, x, 1.0));
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = ((0.25 - t_0) / (0.125 + (0.125 / (fma(x, x, 1.0) * t_1)))) * ((1.0 / (1.0 + sqrt((0.5 + (0.5 / t_1))))) / (1.0 / ((0.25 + t_0) - (0.25 / t_1))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.25 / fma(x, x, 1.0))
	t_1 = sqrt(fma(x, x, 1.0))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(Float64(Float64(0.25 - t_0) / Float64(0.125 + Float64(0.125 / Float64(fma(x, x, 1.0) * t_1)))) * Float64(Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / t_1))))) / Float64(1.0 / Float64(Float64(0.25 + t_0) - Float64(0.25 / t_1)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 - t$95$0), $MachinePrecision] / N[(0.125 + N[(0.125 / N[(N[(x * x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(0.25 + t$95$0), $MachinePrecision] - N[(0.25 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 54.2%

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied egg-rr 99.9%

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

      1. frac-2neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. neg-mul-1 N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   6. Applied egg-rr 99.9%

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

   8. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{\mathsf{fma}\left(x, x, 1\right)}}{1 + \sqrt{t\_0}} \cdot \frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0))))))
   (if (<= (hypot 1.0 x) 1.02)
     (*
      (* x x)
      (fma
       (* x x)
       (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
       0.125))
     (* (/ (+ 0.25 (/ -0.25 (fma x x 1.0))) (+ 1.0 (sqrt t_0))) (/ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / sqrt(fma(x, x, 1.0)));
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = ((0.25 + (-0.25 / fma(x, x, 1.0))) / (1.0 + sqrt(t_0))) * (1.0 / t_0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0))))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(Float64(Float64(0.25 + Float64(-0.25 / fma(x, x, 1.0))) / Float64(1.0 + sqrt(t_0))) * Float64(1.0 / t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 + N[(-0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 54.2%

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied egg-rr 99.9%

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

      1. frac-2neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. neg-mul-1 N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   6. Applied egg-rr 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.125 + \frac{0.125}{\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}}} \cdot \frac{\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}}{\frac{1}{\left(0.25 + \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\right) - \frac{0.25}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}} \]
   9. Step-by-step derivation

      1. frac-times N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

   10. Applied egg-rr 99.9%

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

Alternative 3: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}, 0.5, -0.5\right)}{-1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.02)
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
     0.125))
   (/
    (fma (sqrt (/ 1.0 (fma x x 1.0))) 0.5 -0.5)
    (- -1.0 (sqrt (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0)))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = fma(sqrt((1.0 / fma(x, x, 1.0))), 0.5, -0.5) / (-1.0 - sqrt((0.5 + (0.5 / sqrt(fma(x, x, 1.0))))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(fma(sqrt(Float64(1.0 / fma(x, x, 1.0))), 0.5, -0.5) / Float64(-1.0 - sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0)))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(1.0 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + -0.5), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 54.2%

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. sqrt-div N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. accelerator-lowering-fma.f64 99.9

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(x, x, 1\right)}}, 0.5, -0.5\right)}{-1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (sqrt (fma x x 1.0)))))
   (if (<= (hypot 1.0 x) 1.02)
     (*
      (* x x)
      (fma
       (* x x)
       (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
       0.125))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / sqrt(fma(x, x, 1.0));
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 / sqrt(fma(x, x, 1.0)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 54.2%

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied egg-rr 99.9%

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

      1. frac-2neg N/A

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

      2. +-commutative N/A

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

      3. distribute-neg-in N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. neg-mul-1 N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

   6. Applied egg-rr 99.9%

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

Alternative 5: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.02)
   (*
    (* x x)
    (fma
     (* x x)
     (fma (* x x) (fma (* x x) -0.056243896484375 0.0673828125) -0.0859375)
     0.125))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (sqrt (fma x x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.02) {
		tmp = (x * x) * fma((x * x), fma((x * x), fma((x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / sqrt(fma(x, x, 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.02)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), -0.056243896484375, 0.0673828125), -0.0859375), 0.125));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / sqrt(fma(x, x, 1.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.02], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.02

   1. Initial program 54.2%

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 100.0

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

   7. Simplified 100.0%

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

   if 1.02 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 98.4

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

   4. Applied egg-rr 98.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.02:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.056243896484375 + 0.0673828125), $MachinePrecision] + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.5

   1. Initial program 54.9%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 99.0

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

   7. Simplified 99.0%

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

   if 1.5 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 97.7

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 97.6

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

   10. Simplified 97.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.056243896484375, 0.0673828125\right), -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + \frac{-0.5}{x}}{-1 - \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0673828125 + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.5

   1. Initial program 54.9%

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

   4. Applied egg-rr 55.0%

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

   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 98.8

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

   7. Simplified 98.8%

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

   if 1.5 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. --lowering--.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 97.7

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 97.6

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

   10. Simplified 97.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0673828125, -0.0859375\right), 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 + \frac{-0.5}{x}}{-1 - \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.0673828125 + -0.0859375), $MachinePrecision] + 0.125), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.5

   1. Initial program 54.9%

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

   4. Applied egg-rr 55.0%

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

   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 98.8

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

   7. Simplified 98.8%

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

   if 1.5 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.0%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 97.5

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

   7. Applied egg-rr 97.5%

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

Alternative 9: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.5)
   (* (* x x) (fma (* x x) -0.0859375 0.125))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.5

   1. Initial program 54.9%

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

   4. Applied egg-rr 55.0%

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

   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.5

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

   7. Simplified 98.5%

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

   if 1.5 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.0%

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

      1. flip-- N/A

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

      2. metadata-eval N/A

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

      3. rem-square-sqrt N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 97.5

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

   7. Applied egg-rr 97.5%

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

Alternative 10: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.5:\\ \;\;\;\;\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, -0.0859375, 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.5)
   (* (* x x) (fma (* x x) -0.0859375 0.125))
   (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.5) {
		tmp = (x * x) * fma((x * x), -0.0859375, 0.125);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.5)
		tmp = Float64(Float64(x * x) * fma(Float64(x * x), -0.0859375, 0.125));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.5], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0859375 + 0.125), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.5

   1. Initial program 54.9%

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

   4. Applied egg-rr 55.0%

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

   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. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 98.5

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

   7. Simplified 98.5%

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

   if 1.5 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.0%

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

Alternative 11: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2%

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.6

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

   7. Simplified 97.6%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 96.6%

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

4. Final simplification 97.1%

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

Alternative 12: 60.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* (* x x) 0.125) 0.25))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * 0.125;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * 0.125;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (x * x) * 0.125
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(x * x) * 0.125);
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (x * x) * 0.125;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(x * x), $MachinePrecision] * 0.125), $MachinePrecision], 0.25]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2%

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

   4. Applied egg-rr 55.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 97.6

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

   7. Simplified 97.6%

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

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

   1. Initial program 98.5%

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

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 22.7%

      \[\leadsto \color{blue}{0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 13.6% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 77.4%

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

4. Applied egg-rr 52.7%

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

5. Taylor expanded in x around 0

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

7. Simplified 13.8%

   \[\leadsto \color{blue}{0.25} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))