Given's Rotation SVD example, simplified

Percentage Accurate: 75.6% → 99.9%

Time: 9.7s

Alternatives: 17

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: 75.6% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \sqrt{t\_0}\\ t_2 := \mathsf{fma}\left(-1, t\_1, -1\right)\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{t\_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{t\_0}{t\_2}\right)}^{2} - {t\_2}^{-2}}{\left(t\_0 + 1\right) \cdot {\left(-1 - t\_1\right)}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x))))
        (t_1 (sqrt t_0))
        (t_2 (fma -1.0 t_1 -1.0)))
   (if (<= (hypot 1.0 x) 1.1)
     (/
      (- (* (* (fma (fma -0.15625 (* x x) 0.1875) (* x x) -0.25) x) x))
      (+ t_1 1.0))
     (/
      (- (pow (/ t_0 t_2) 2.0) (pow t_2 -2.0))
      (* (+ t_0 1.0) (pow (- -1.0 t_1) -1.0))))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double t_1 = sqrt(t_0);
	double t_2 = fma(-1.0, t_1, -1.0);
	double tmp;
	if (hypot(1.0, x) <= 1.1) {
		tmp = -((fma(fma(-0.15625, (x * x), 0.1875), (x * x), -0.25) * x) * x) / (t_1 + 1.0);
	} else {
		tmp = (pow((t_0 / t_2), 2.0) - pow(t_2, -2.0)) / ((t_0 + 1.0) * pow((-1.0 - t_1), -1.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	t_1 = sqrt(t_0)
	t_2 = fma(-1.0, t_1, -1.0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.1)
		tmp = Float64(Float64(-Float64(Float64(fma(fma(-0.15625, Float64(x * x), 0.1875), Float64(x * x), -0.25) * x) * x)) / Float64(t_1 + 1.0));
	else
		tmp = Float64(Float64((Float64(t_0 / t_2) ^ 2.0) - (t_2 ^ -2.0)) / Float64(Float64(t_0 + 1.0) * (Float64(-1.0 - t_1) ^ -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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * t$95$1 + -1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.1], N[((-N[(N[(N[(N[(-0.15625 * N[(x * x), $MachinePrecision] + 0.1875), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]) / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(t$95$0 / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[t$95$2, -2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 + 1.0), $MachinePrecision] * N[Power[N[(-1.0 - t$95$1), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.1000000000000001

   1. Initial program 54.9%

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2}}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right)} \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\color{blue}{\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\frac{-5}{32} \cdot {x}^{2} + \frac{3}{16}}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5}{32}, {x}^{2}, \frac{3}{16}\right)}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), \color{blue}{x \cdot x}, -0.25\right) \cdot x\right) \cdot x}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   if 1.1000000000000001 < (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. div-sub N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. unpow-1 N/A

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

      5. lift-pow.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}\right)}^{2} - {\left(\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)\right)}^{-2}}{\left(\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) + 1\right) \cdot {\left(-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \sqrt{t\_0} + 1\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_1}{1 - t\_0}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))) (t_1 (+ (sqrt t_0) 1.0)))
   (if (<= (hypot 1.0 x) 1.1)
     (/ (- (* (* (fma (fma -0.15625 (* x x) 0.1875) (* x x) -0.25) x) x)) t_1)
     (pow (/ t_1 (- 1.0 t_0)) -1.0))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double t_1 = sqrt(t_0) + 1.0;
	double tmp;
	if (hypot(1.0, x) <= 1.1) {
		tmp = -((fma(fma(-0.15625, (x * x), 0.1875), (x * x), -0.25) * x) * x) / t_1;
	} else {
		tmp = pow((t_1 / (1.0 - t_0)), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	t_1 = Float64(sqrt(t_0) + 1.0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.1)
		tmp = Float64(Float64(-Float64(Float64(fma(fma(-0.15625, Float64(x * x), 0.1875), Float64(x * x), -0.25) * x) * x)) / t_1);
	else
		tmp = Float64(t_1 / Float64(1.0 - 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]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.1], N[((-N[(N[(N[(N[(-0.15625 * N[(x * x), $MachinePrecision] + 0.1875), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]) / t$95$1), $MachinePrecision], N[Power[N[(t$95$1 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.1000000000000001

   1. Initial program 54.9%

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2}}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right)} \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\color{blue}{\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\frac{-5}{32} \cdot {x}^{2} + \frac{3}{16}}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5}{32}, {x}^{2}, \frac{3}{16}\right)}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), \color{blue}{x \cdot x}, -0.25\right) \cdot x\right) \cdot x}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   if 1.1000000000000001 < (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. div-sub N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}{1 - \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{t\_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 1}{-1 - t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x)))) (t_1 (sqrt t_0)))
   (if (<= (hypot 1.0 x) 1.1)
     (/
      (- (* (* (fma (fma -0.15625 (* x x) 0.1875) (* x x) -0.25) x) x))
      (+ t_1 1.0))
     (/ (- t_0 1.0) (- -1.0 t_1)))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double t_1 = sqrt(t_0);
	double tmp;
	if (hypot(1.0, x) <= 1.1) {
		tmp = -((fma(fma(-0.15625, (x * x), 0.1875), (x * x), -0.25) * x) * x) / (t_1 + 1.0);
	} else {
		tmp = (t_0 - 1.0) / (-1.0 - t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.1)
		tmp = Float64(Float64(-Float64(Float64(fma(fma(-0.15625, Float64(x * x), 0.1875), Float64(x * x), -0.25) * x) * x)) / Float64(t_1 + 1.0));
	else
		tmp = Float64(Float64(t_0 - 1.0) / Float64(-1.0 - t_1));
	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]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.1], N[((-N[(N[(N[(N[(-0.15625 * N[(x * x), $MachinePrecision] + 0.1875), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]) / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.1000000000000001

   1. Initial program 54.9%

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2}}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right)} \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\color{blue}{\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\frac{-5}{32} \cdot {x}^{2} + \frac{3}{16}}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5}{32}, {x}^{2}, \frac{3}{16}\right)}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), \color{blue}{x \cdot x}, -0.25\right) \cdot x\right) \cdot x}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   if 1.1000000000000001 < (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. div-sub N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

   6. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\sqrt{0.5 - t\_0} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\left(1 - t\_0\right) - 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.1)
     (/
      (- (* (* (fma (fma -0.15625 (* x x) 0.1875) (* x x) -0.25) x) x))
      (+ (sqrt (- 0.5 t_0)) 1.0))
     (- 1.0 (sqrt (- (- 1.0 t_0) 0.5))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.1) {
		tmp = -((fma(fma(-0.15625, (x * x), 0.1875), (x * x), -0.25) * x) * x) / (sqrt((0.5 - t_0)) + 1.0);
	} else {
		tmp = 1.0 - sqrt(((1.0 - t_0) - 0.5));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.1)
		tmp = Float64(Float64(-Float64(Float64(fma(fma(-0.15625, Float64(x * x), 0.1875), Float64(x * x), -0.25) * x) * x)) / Float64(sqrt(Float64(0.5 - t_0)) + 1.0));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(1.0 - t_0) - 0.5)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.1], N[((-N[(N[(N[(N[(-0.15625 * N[(x * x), $MachinePrecision] + 0.1875), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]) / N[(N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.1000000000000001

   1. Initial program 54.9%

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot {x}^{2}}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      2. unpow2 N/A

         \[\leadsto \frac{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot \color{blue}{\left(x \cdot x\right)}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x}}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) - \frac{1}{4}\right) \cdot x\right)} \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      6. sub-neg N/A

         \[\leadsto \frac{\left(\color{blue}{\left({x}^{2} \cdot \left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\left(\color{blue}{\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{4}}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(\frac{3}{16} + \frac{-5}{32} \cdot {x}^{2}, {x}^{2}, \frac{-1}{4}\right)} \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\frac{-5}{32} \cdot {x}^{2} + \frac{3}{16}}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-5}{32}, {x}^{2}, \frac{3}{16}\right)}, {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      12. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, \color{blue}{x \cdot x}, \frac{3}{16}\right), {x}^{2}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      14. unpow2 N/A

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-5}{32}, x \cdot x, \frac{3}{16}\right), \color{blue}{x \cdot x}, \frac{-1}{4}\right) \cdot x\right) \cdot x}{\left(-\sqrt{\frac{1}{2} - \frac{\frac{-1}{2}}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

      15. lower-*.f64 100.0

          \[\leadsto \frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), \color{blue}{x \cdot x}, -0.25\right) \cdot x\right) \cdot x}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   if 1.1000000000000001 < (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 98.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.1:\\ \;\;\;\;\frac{-\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.15625, x \cdot x, 0.1875\right), x \cdot x, -0.25\right) \cdot x\right) \cdot x}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\left(1 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x) 0.5)))
   (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)
     (pow (/ (+ (sqrt t_0) 1.0) (- 1.0 t_0)) -1.0))))

C

double code(double x) {
	double t_0 = (0.5 / x) + 0.5;
	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 = pow(((sqrt(t_0) + 1.0) / (1.0 - t_0)), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 / x) + 0.5)
	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(Float64(sqrt(t_0) + 1.0) / Float64(1.0 - t_0)) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]}, 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[Power[N[(N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   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 99.5%

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

   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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

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

      3. clear-num N/A

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

   7. Applied rewrites 98.0%

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

4. Final simplification 98.9%

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

Alternative 6: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.005:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (pow (hypot 1.0 x) -1.0) 0.005)
   (- 1.0 (sqrt (+ (/ 0.5 x) 0.5)))
   (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)))

C

double code(double x) {
	double tmp;
	if (pow(hypot(1.0, x), -1.0) <= 0.005) {
		tmp = 1.0 - sqrt(((0.5 / x) + 0.5));
	} else {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((hypot(1.0, x) ^ -1.0) <= 0.005)
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x) + 0.5)));
	else
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -1.0], $MachinePrecision], 0.005], N[(1.0 - N[Sqrt[N[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0673828125 * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 0.0050000000000000001

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   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 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.005:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.1)
   (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
   (- 1.0 (sqrt (- (- 1.0 (/ -0.5 (hypot 1.0 x))) 0.5)))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.1], 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[(1.0 - N[Sqrt[N[(N[(1.0 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.1000000000000001

   1. Initial program 54.9%

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in x around 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 100.0

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

   7. Applied rewrites 100.0%

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

   if 1.1000000000000001 < (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 98.4%

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

Alternative 8: 99.1% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.1)
   (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
   (- 1.0 (sqrt (- 0.5 (/ -0.5 (hypot 1.0 x)))))))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.1], 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[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.1000000000000001

   1. Initial program 54.9%

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in x around 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 100.0

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

   7. Applied rewrites 100.0%

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

   if 1.1000000000000001 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. inv-pow N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-pow N/A

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

      16. pow2 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. inv-pow N/A

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

      23. lift-/.f64 N/A

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

   4. Applied rewrites 98.4%

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

Alternative 9: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.005:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (pow (hypot 1.0 x) -1.0) 0.005)
   (- 1.0 (sqrt (+ (/ 0.5 x) 0.5)))
   (* (* (fma -0.0859375 (* x x) 0.125) x) x)))

C

double code(double x) {
	double tmp;
	if (pow(hypot(1.0, x), -1.0) <= 0.005) {
		tmp = 1.0 - sqrt(((0.5 / x) + 0.5));
	} else {
		tmp = (fma(-0.0859375, (x * x), 0.125) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((hypot(1.0, x) ^ -1.0) <= 0.005)
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x) + 0.5)));
	else
		tmp = Float64(Float64(fma(-0.0859375, Float64(x * x), 0.125) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -1.0], $MachinePrecision], 0.005], N[(1.0 - N[Sqrt[N[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.0859375 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 0.0050000000000000001

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if 0.0050000000000000001 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   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%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.005:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 0.8× speedup

Math

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

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)
   (- 1.0 (* (sqrt (+ (/ (- 1.0 (/ 0.5 (* x x))) x) 1.0)) (sqrt 0.5)))))

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 = 1.0 - (sqrt((((1.0 - (0.5 / (x * x))) / x) + 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(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(1.0 - Float64(sqrt(Float64(Float64(Float64(1.0 - Float64(0.5 / Float64(x * x))) / x) + 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[(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[(1.0 - N[(N[Sqrt[N[(N[(N[(1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   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 99.5%

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

   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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 96.8

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

   5. Applied rewrites 96.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.8

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

   10. Applied rewrites 96.8%

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

Alternative 11: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{\frac{0.25}{x \cdot x} - 0.5}{x}}\\ \end{array} \end{array} \]

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)
   (- 1.0 (sqrt (- 0.5 (/ (- (/ 0.25 (* x x)) 0.5) x))))))

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 = 1.0 - sqrt((0.5 - (((0.25 / (x * x)) - 0.5) / x)));
	}
	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(1.0 - sqrt(Float64(0.5 - Float64(Float64(Float64(0.25 / Float64(x * x)) - 0.5) / x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-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[(1.0 - N[Sqrt[N[(0.5 - N[(N[(N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   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 99.5%

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

   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
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. inv-pow N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-pow N/A

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

      16. pow2 N/A

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

      17. sqr-neg N/A

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

      18. pow-prod-down N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. inv-pow N/A

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

      23. lift-/.f64 N/A

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 96.8

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

   7. Applied rewrites 96.8%

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

Alternative 12: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.2%

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

   4. Applied rewrites 55.3%

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

   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 99.5%

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

   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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 13: 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 55.2%

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

   4. Applied rewrites 55.3%

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

   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 97.4

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

   7. Applied rewrites 97.4%

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

Alternative 14: 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 55.2%

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

   4. Applied rewrites 55.3%

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

   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 95.9%

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

Alternative 15: 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 55.2%

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

   4. Applied rewrites 55.3%

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

   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 98.7

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

   7. Applied rewrites 98.7%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.9%

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

Alternative 16: 52.6% 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 73.8%

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

4. Applied rewrites 74.5%

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

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 58.1

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

7. Applied rewrites 58.1%

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

Alternative 17: 28.2% accurate, 33.5× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 1.0))

C

double code(double x) {
	return 1.0 - 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 - 1.0;
}

Python

def code(x):
	return 1.0 - 1.0

Julia

function code(x)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 73.8%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. inv-pow N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

   16. pow2 N/A

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

   17. sqr-neg N/A

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

   18. pow-prod-down N/A

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

   19. pow-sqr N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. inv-pow N/A

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

   23. lift-/.f64 N/A

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

4. Applied rewrites 73.8%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 33.0

      \[\leadsto 1 - \mathsf{fma}\left(-0.125, \color{blue}{x \cdot x}, 1\right) \]

7. Applied rewrites 33.0%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(-0.125, x \cdot x, 1\right)} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 31.8%

    \[\leadsto 1 - \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024315 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))