Given's Rotation SVD example, simplified

Percentage Accurate: 75.7% → 100.0%

Time: 5.5s

Alternatives: 13

Speedup: 6.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.7% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left({\left(\mathsf{fma}\left(x\_m, x\_m, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)\\ \mathbf{if}\;x\_m \leq 0.029:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (pow (fma x_m x_m 1.0) -0.5) 0.5 0.5)))
   (if (<= x_m 0.029)
     (*
      (*
       (fma
        (-
         (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
         0.0859375)
        (* x_m x_m)
        0.125)
       x_m)
      x_m)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(pow(fma(x_m, x_m, 1.0), -0.5), 0.5, 0.5);
	double tmp;
	if (x_m <= 0.029) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma((fma(x_m, x_m, 1.0) ^ -0.5), 0.5, 0.5)
	tmp = 0.0
	if (x_m <= 0.029)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Power[N[(x$95$m * x$95$m + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 0.029], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0290000000000000015

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 0.0290000000000000015 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. inv-pow N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. sqrt-pow2 N/A

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

      10. metadata-eval N/A

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

      11. lift-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.82:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \mathsf{fma}\left({\left(\mathsf{fma}\left(x\_m, x\_m, 1\right)\right)}^{-0.5}, 0.5, 0.5\right)}{\sqrt{\mathsf{fma}\left(\frac{1 - \frac{0.5}{x\_m \cdot x\_m}}{x\_m}, 0.5, 0.5\right)} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.82)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
       0.0859375)
      (* x_m x_m)
      0.125)
     x_m)
    x_m)
   (/
    (- 1.0 (fma (pow (fma x_m x_m 1.0) -0.5) 0.5 0.5))
    (+ (sqrt (fma (/ (- 1.0 (/ 0.5 (* x_m x_m))) x_m) 0.5 0.5)) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.82) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (1.0 - fma(pow(fma(x_m, x_m, 1.0), -0.5), 0.5, 0.5)) / (sqrt(fma(((1.0 - (0.5 / (x_m * x_m))) / x_m), 0.5, 0.5)) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.82)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(1.0 - fma((fma(x_m, x_m, 1.0) ^ -0.5), 0.5, 0.5)) / Float64(sqrt(fma(Float64(Float64(1.0 - Float64(0.5 / Float64(x_m * x_m))) / x_m), 0.5, 0.5)) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.82], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 - N[(N[Power[N[(x$95$m * x$95$m + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(1.0 - N[(0.5 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 0.819999999999999951 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. pow1/2 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan-rev N/A

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

      4. inv-pow N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. sqrt-pow2 N/A

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

      10. metadata-eval N/A

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

      11. lift-pow.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. associate-*r/ N/A

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

       4. metadata-eval N/A

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

       5. lower-/.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 99.3

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

   11. Applied rewrites 99.3%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} - -0.5\\ t_1 := \sqrt{t\_0} + 1\\ \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1} - \frac{t\_0}{t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ 0.5 x_m) -0.5)) (t_1 (+ (sqrt t_0) 1.0)))
   (if (<= x_m 1.1)
     (*
      (*
       (fma
        (-
         (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
         0.0859375)
        (* x_m x_m)
        0.125)
       x_m)
      x_m)
     (- (/ 1.0 t_1) (/ t_0 t_1)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) - -0.5;
	double t_1 = sqrt(t_0) + 1.0;
	double tmp;
	if (x_m <= 1.1) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = (1.0 / t_1) - (t_0 / t_1);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) - -0.5)
	t_1 = Float64(sqrt(t_0) + 1.0)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(Float64(1.0 / t_1) - Float64(t_0 / t_1));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(1.0 / t$95$1), $MachinePrecision] - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 99.0%

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

Alternative 4: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
       0.0859375)
      (* x_m x_m)
      0.125)
     x_m)
    x_m)
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.032)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.032], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 0.032000000000000001 < x

   1. Initial program 98.5%

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

      1. lift-hypot.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 5: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.88:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\frac{0.5}{x\_m} + x\_m}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
       0.0859375)
      (* x_m x_m)
      0.125)
     x_m)
    x_m)
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (+ (/ 0.5 x_m) x_m))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / ((0.5 / x_m) + x_m)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / Float64(Float64(0.5 / x_m) + x_m))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[(N[(0.5 / x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.880000000000000004

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 0.880000000000000004 < 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 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 94.9

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*l/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 97.8

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

   8. Applied rewrites 97.8%

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

Alternative 6: 98.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
       0.0859375)
      (* x_m x_m)
      0.125)
     x_m)
    x_m)
   (- 1.0 (sqrt (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt((((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.05], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 1.05000000000000004 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. div-sub N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. fp-cancel-sub-sign N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

   5. Applied rewrites 97.8%

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

Alternative 7: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-0.056243896484375, x\_m \cdot x\_m, 0.0673828125\right) \cdot x\_m\right) \cdot x\_m - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot x\_m\right) \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.1)
   (*
    (*
     (fma
      (-
       (* (* (fma -0.056243896484375 (* x_m x_m) 0.0673828125) x_m) x_m)
       0.0859375)
      (* x_m x_m)
      0.125)
     x_m)
    x_m)
   (- 1.0 (sqrt (- (/ 0.5 x_m) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.1) {
		tmp = (fma((((fma(-0.056243896484375, (x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), (x_m * x_m), 0.125) * x_m) * x_m;
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.1)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.056243896484375, Float64(x_m * x_m), 0.0673828125) * x_m) * x_m) - 0.0859375), Float64(x_m * x_m), 0.125) * x_m) * x_m);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.1], N[(N[(N[(N[(N[(N[(N[(-0.056243896484375 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 68.0

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

   10. Applied rewrites 68.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 68.1%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 8: 98.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(0.0673828125 \cdot \left(x\_m \cdot x\_m\right) - 0.0859375, x\_m \cdot x\_m, 0.125\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.2)
   (*
    (fma (- (* 0.0673828125 (* x_m x_m)) 0.0859375) (* x_m x_m) 0.125)
    (* x_m x_m))
   (- 1.0 (sqrt (- (/ 0.5 x_m) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.2) {
		tmp = fma(((0.0673828125 * (x_m * x_m)) - 0.0859375), (x_m * x_m), 0.125) * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(fma(Float64(Float64(0.0673828125 * Float64(x_m * x_m)) - 0.0859375), Float64(x_m * x_m), 0.125) * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.2], N[(N[(N[(N[(0.0673828125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 68.9

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

   10. Applied rewrites 68.9%

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

   if 1.19999999999999996 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 9: 98.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 10: 98.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 97.5

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

   5. Applied rewrites 97.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 96.4

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

   10. Applied rewrites 96.4%

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

Alternative 11: 97.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if 1.1000000000000001 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.9%

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

Alternative 12: 97.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.52:\\ \;\;\;\;0.125 \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.52) (* 0.125 (* x_m x_m)) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.52) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.52d0) then
        tmp = 0.125d0 * (x_m * x_m)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.52) {
		tmp = 0.125 * (x_m * x_m);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.52:
		tmp = 0.125 * (x_m * x_m)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.52)
		tmp = Float64(0.125 * Float64(x_m * x_m));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.52)
		tmp = 0.125 * (x_m * x_m);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.52], N[(0.125 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 65.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 67.8

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

   10. Applied rewrites 67.8%

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

   if 1.52 < x

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 94.9%

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

Alternative 13: 51.7% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 72.1

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

5. Applied rewrites 72.1%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f64 N/A

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

7. Applied rewrites 72.1%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 52.0

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

10. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))