Given's Rotation SVD example, simplified

Percentage Accurate: 75.7% → 99.9%

Time: 8.6s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \sqrt{0.5 - t\_0}\\ t_2 := \left(1.5 + t\_0\right) \cdot {\left(-1 - t\_1\right)}^{3}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 \cdot {\left(\mathsf{fma}\left(t\_1, -0.5, -0.5\right)\right)}^{2} - {\left(\left(t\_1 + 1\right) \cdot \left(-1 - t\_0\right)\right)}^{2} \cdot t\_2}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x)))
        (t_1 (sqrt (- 0.5 t_0)))
        (t_2 (* (+ 1.5 t_0) (pow (- -1.0 t_1) 3.0))))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/
      (-
       (* t_2 (pow (fma t_1 -0.5 -0.5) 2.0))
       (* (pow (* (+ t_1 1.0) (- -1.0 t_0)) 2.0) t_2))
      (* t_2 t_2)))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = sqrt((0.5 - t_0));
	double t_2 = (1.5 + t_0) * pow((-1.0 - t_1), 3.0);
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = ((t_2 * pow(fma(t_1, -0.5, -0.5), 2.0)) - (pow(((t_1 + 1.0) * (-1.0 - t_0)), 2.0) * t_2)) / (t_2 * t_2);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = sqrt(Float64(0.5 - t_0))
	t_2 = Float64(Float64(1.5 + t_0) * (Float64(-1.0 - t_1) ^ 3.0))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(t_2 * (fma(t_1, -0.5, -0.5) ^ 2.0)) - Float64((Float64(Float64(t_1 + 1.0) * Float64(-1.0 - t_0)) ^ 2.0) * t_2)) / Float64(t_2 * t_2));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.5 + t$95$0), $MachinePrecision] * N[Power[N[(-1.0 - t$95$1), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 * N[Power[N[(t$95$1 * -0.5 + -0.5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[(N[(t$95$1 + 1.0), $MachinePrecision] * N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 48.4%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. div-sub N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right) - \mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right) \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}{{\left(\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)\right)}^{2}}} \]

   8. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -0.5, -0.5\right)\right)}^{2} - {\left(\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right) \cdot \left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)\right)}^{2}}{{\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{2} \cdot \left(\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

   10. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot {\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right) \cdot {\left(\mathsf{fma}\left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -0.5, -0.5\right)\right)}^{2} - {\left(\left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1\right) \cdot \left(-1 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{2} \cdot \left(\left(1.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot {\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right)}{\left(\left(1.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot {\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right) \cdot \left(\left(1.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot {\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{3} - 1}{\left(\left({t\_0}^{2} + 1\right) + t\_0\right) \cdot \mathsf{fma}\left(-1, \sqrt{t\_0}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/
      (- (pow t_0 3.0) 1.0)
      (* (+ (+ (pow t_0 2.0) 1.0) t_0) (fma -1.0 (sqrt t_0) -1.0))))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (pow(t_0, 3.0) - 1.0) / (((pow(t_0, 2.0) + 1.0) + t_0) * fma(-1.0, sqrt(t_0), -1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64((t_0 ^ 3.0) - 1.0) / Float64(Float64(Float64((t_0 ^ 2.0) + 1.0) + t_0) * fma(-1.0, sqrt(t_0), -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] + 1.0), $MachinePrecision] + t$95$0), $MachinePrecision] * N[(-1.0 * N[Sqrt[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 48.4%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip3-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} - 1}{\left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right) \cdot \mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \mathsf{fma}\left(-1, \sqrt{0.5 - t\_0}, -1\right)\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot 0.5 - t\_1 \cdot \left(t\_0 + 1\right)}{{t\_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))) (t_1 (fma -1.0 (sqrt (- 0.5 t_0)) -1.0)))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/ (- (* t_1 0.5) (* t_1 (+ t_0 1.0))) (pow t_1 2.0)))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = fma(-1.0, sqrt((0.5 - t_0)), -1.0);
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = ((t_1 * 0.5) - (t_1 * (t_0 + 1.0))) / pow(t_1, 2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = fma(-1.0, sqrt(Float64(0.5 - t_0)), -1.0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(t_1 * 0.5) - Float64(t_1 * Float64(t_0 + 1.0))) / (t_1 ^ 2.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * 0.5), $MachinePrecision] - N[(t$95$1 * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 48.4%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. div-sub N/A

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

      6. frac-sub N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right) \cdot 0.5 - \mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right) \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}{{\left(\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {t\_1}^{1.5}}{\left(0.5 - \left(t\_0 - 1\right)\right) + \sqrt{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))) (t_1 (- 0.5 t_0)))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/ (- 1.0 (pow t_1 1.5)) (+ (- 0.5 (- t_0 1.0)) (sqrt t_1))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (1.0 - pow(t_1, 1.5)) / ((0.5 - (t_0 - 1.0)) + sqrt(t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 1.5)) / Float64(Float64(0.5 - Float64(t_0 - 1.0)) + sqrt(t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 1.5], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 - N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] + N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 48.4%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 99.9%

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

Alternative 5: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{\sqrt{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0002)
   (*
    (fma
     (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
     (* x x)
     0.125)
    (* x x))
   (/
    (- 1.0 (- 0.5 (/ -0.5 (hypot 1.0 x))))
    (+
     (sqrt (/ (- 0.25 (/ 0.25 (fma x x 1.0))) (- 0.5 (/ 0.5 (hypot 1.0 x)))))
     1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (1.0 - (0.5 - (-0.5 / hypot(1.0, x)))) / (sqrt(((0.25 - (0.25 / fma(x, x, 1.0))) / (0.5 - (0.5 / hypot(1.0, x))))) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))) / Float64(sqrt(Float64(Float64(0.25 - Float64(0.25 / fma(x, x, 1.0))) / Float64(0.5 - Float64(0.5 / hypot(1.0, x))))) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(N[(0.25 - N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 48.4%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. lift-hypot.f64 N/A

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

      5. metadata-eval N/A

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

      6. cancel-sign-sub-inv N/A

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

      7. metadata-eval N/A

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

      8. flip-+ N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{\sqrt{\frac{0.25 - \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}}{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0002)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0002) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0002)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0002], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.0002

   1. Initial program 48.4%

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.0002 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

   4. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 99.9%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. 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 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 8: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in x around 0

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

      1. *-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. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. 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 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 9: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. 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 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 10: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{1 + \sqrt{\frac{1}{2}}}} \]
   6. 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 98.6

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

   7. Applied rewrites 98.6%

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

Alternative 11: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.4

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

   7. Applied rewrites 99.4%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.1%

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

Alternative 12: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.7%

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

   4. Applied rewrites 48.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.1%

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

Alternative 13: 51.5% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

4. Applied rewrites 77.8%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
6. 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 45.2

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

7. Applied rewrites 45.2%

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

Reproduce

herbie shell --seed 2024277 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))