Given's Rotation SVD example, simplified

Percentage Accurate: 75.4% → 99.9%

Time: 9.2s

Alternatives: 12

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: 75.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ t_2 := {t\_1}^{2}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t\_1}^{3} \cdot \left(\left(t\_2 + 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right), {\left({\left(t\_2 + 1\right)}^{2} - t\_2\right)}^{-1}, \frac{-1}{\left(1.5 + t\_2\right) - t\_0}\right)}{-1 - \sqrt{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))) (t_1 (- 0.5 t_0)) (t_2 (pow t_1 2.0)))
   (if (<= (hypot 1.0 x) 1.0005)
     (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
     (/
      (fma
       (* (pow t_1 3.0) (- (+ t_2 0.5) (/ 0.5 (hypot 1.0 x))))
       (pow (- (pow (+ t_2 1.0) 2.0) t_2) -1.0)
       (/ -1.0 (- (+ 1.5 t_2) t_0)))
      (- -1.0 (sqrt t_1))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double t_2 = pow(t_1, 2.0);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = fma((pow(t_1, 3.0) * ((t_2 + 0.5) - (0.5 / hypot(1.0, x)))), pow((pow((t_2 + 1.0), 2.0) - t_2), -1.0), (-1.0 / ((1.5 + t_2) - t_0))) / (-1.0 - sqrt(t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	t_2 = t_1 ^ 2.0
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(fma(Float64((t_1 ^ 3.0) * Float64(Float64(t_2 + 0.5) - Float64(0.5 / hypot(1.0, x)))), (Float64((Float64(t_2 + 1.0) ^ 2.0) - t_2) ^ -1.0), Float64(-1.0 / Float64(Float64(1.5 + t_2) - t_0))) / Float64(-1.0 - sqrt(t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], 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[(N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] * N[(N[(t$95$2 + 0.5), $MachinePrecision] - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[Power[N[(t$95$2 + 1.0), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision], -1.0], $MachinePrecision] + N[(-1.0 / N[(N[(1.5 + t$95$2), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 54.7%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} - {\left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{-1}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right)}^{2} - {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}}, \left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}, \frac{-1}{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1.5\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

   10. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3} \cdot \left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right), {\left({\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right)}^{2} - {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right)}^{-1}, \frac{-1}{\left(1.5 + {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ t_2 := {t\_1}^{2}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t\_1}^{3}, \frac{\left(t\_2 + 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{{\left(t\_2 + 1\right)}^{2} - t\_2}, \frac{-1}{\left(1.5 + t\_2\right) - t\_0}\right)}{-1 - \sqrt{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))) (t_1 (- 0.5 t_0)) (t_2 (pow t_1 2.0)))
   (if (<= (hypot 1.0 x) 1.0005)
     (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
     (/
      (fma
       (pow t_1 3.0)
       (/ (- (+ t_2 0.5) (/ 0.5 (hypot 1.0 x))) (- (pow (+ t_2 1.0) 2.0) t_2))
       (/ -1.0 (- (+ 1.5 t_2) t_0)))
      (- -1.0 (sqrt t_1))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double t_2 = pow(t_1, 2.0);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = fma(pow(t_1, 3.0), (((t_2 + 0.5) - (0.5 / hypot(1.0, x))) / (pow((t_2 + 1.0), 2.0) - t_2)), (-1.0 / ((1.5 + t_2) - t_0))) / (-1.0 - sqrt(t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	t_2 = t_1 ^ 2.0
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(fma((t_1 ^ 3.0), Float64(Float64(Float64(t_2 + 0.5) - Float64(0.5 / hypot(1.0, x))) / Float64((Float64(t_2 + 1.0) ^ 2.0) - t_2)), Float64(-1.0 / Float64(Float64(1.5 + t_2) - t_0))) / Float64(-1.0 - sqrt(t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], 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[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] * N[(N[(N[(t$95$2 + 0.5), $MachinePrecision] - N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$2 + 1.0), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(1.5 + t$95$2), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 54.7%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} - {\left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{-1}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

   8. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right)}^{2} - {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}}, \left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}, \frac{-1}{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1.5\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   9. Step-by-step derivation

      1. lift-fma.f64 N/A

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

   10. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}, \frac{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right)}^{2} - {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}}, \frac{-1}{\left(1.5 + {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ t_2 := \left(1.5 + {t\_1}^{2}\right) - t\_0\\ t_3 := \mathsf{fma}\left(-1, \sqrt{t\_1}, -1\right)\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_1}^{3}}{t\_2}, {t\_3}^{-1}, \frac{-{t\_2}^{-1}}{t\_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x)))
        (t_1 (- 0.5 t_0))
        (t_2 (- (+ 1.5 (pow t_1 2.0)) t_0))
        (t_3 (fma -1.0 (sqrt t_1) -1.0)))
   (if (<= (hypot 1.0 x) 1.0005)
     (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
     (fma (/ (pow t_1 3.0) t_2) (pow t_3 -1.0) (/ (- (pow t_2 -1.0)) t_3)))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double t_2 = (1.5 + pow(t_1, 2.0)) - t_0;
	double t_3 = fma(-1.0, sqrt(t_1), -1.0);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = fma((pow(t_1, 3.0) / t_2), pow(t_3, -1.0), (-pow(t_2, -1.0) / t_3));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	t_2 = Float64(Float64(1.5 + (t_1 ^ 2.0)) - t_0)
	t_3 = fma(-1.0, sqrt(t_1), -1.0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = fma(Float64((t_1 ^ 3.0) / t_2), (t_3 ^ -1.0), Float64(Float64(-(t_2 ^ -1.0)) / t_3));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.5 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * N[Sqrt[t$95$1], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], 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[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] / t$95$2), $MachinePrecision] * N[Power[t$95$3, -1.0], $MachinePrecision] + N[((-N[Power[t$95$2, -1.0], $MachinePrecision]) / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 54.7%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} - {\left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{-1}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1.5 + {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, {\left(\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)\right)}^{-1}, \frac{-{\left(\left(1.5 + {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{-1}}{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := 0.5 - t\_0\\ t_2 := \left(1.5 + {t\_1}^{2}\right) - t\_0\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{t\_2} + \frac{{t\_1}^{3}}{t\_2}}{-1 - \sqrt{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x)))
        (t_1 (- 0.5 t_0))
        (t_2 (- (+ 1.5 (pow t_1 2.0)) t_0)))
   (if (<= (hypot 1.0 x) 1.0005)
     (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
     (/ (+ (/ -1.0 t_2) (/ (pow t_1 3.0) t_2)) (- -1.0 (sqrt t_1))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double t_1 = 0.5 - t_0;
	double t_2 = (1.5 + pow(t_1, 2.0)) - t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = ((-1.0 / t_2) + (pow(t_1, 3.0) / t_2)) / (-1.0 - sqrt(t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	t_1 = Float64(0.5 - t_0)
	t_2 = Float64(Float64(1.5 + (t_1 ^ 2.0)) - t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(Float64(-1.0 / t_2) + Float64((t_1 ^ 3.0) / t_2)) / Float64(-1.0 - sqrt(t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.5 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], 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[(N[(N[(-1.0 / t$95$2), $MachinePrecision] + N[(N[Power[t$95$1, 3.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 54.7%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. metadata-eval N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} - {\left(\left({\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2} + 1\right) + \left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}^{-1}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\left(1.5 + {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + \frac{{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\left(1.5 + {\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right) - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{-1 - \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ t_1 := \mathsf{fma}\left(-1, \sqrt{t\_0}, -1\right)\\ t_2 := \frac{t\_1}{t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 - t\_2}{t\_2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (hypot 1.0 x))))
        (t_1 (fma -1.0 (sqrt t_0) -1.0))
        (t_2 (/ t_1 t_0)))
   (if (<= (hypot 1.0 x) 1.0005)
     (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
     (/ (- t_1 t_2) (* t_2 t_1)))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / hypot(1.0, x));
	double t_1 = fma(-1.0, sqrt(t_0), -1.0);
	double t_2 = t_1 / t_0;
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = (t_1 - t_2) / (t_2 * t_1);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / hypot(1.0, x)))
	t_1 = fma(-1.0, sqrt(t_0), -1.0)
	t_2 = Float64(t_1 / t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(t_1 - t_2) / Float64(t_2 * 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[(-1.0 * N[Sqrt[t$95$0], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], 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[(N[(t$95$1 - t$95$2), $MachinePrecision] / N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 54.7%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\left(0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 1}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1}} \]
   5. 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. clear-num N/A

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

      5. frac-sub N/A

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

      6. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right) - \frac{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{\frac{\mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \mathsf{fma}\left(-1, \sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_0 - 0.5\right) - -1}{\sqrt{0.5 - t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)
     (/ (- (- t_0 0.5) -1.0) (+ (sqrt (- 0.5 t_0)) 1.0)))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = ((t_0 - 0.5) - -1.0) / (sqrt((0.5 - t_0)) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(Float64(t_0 - 0.5) - -1.0) / Float64(sqrt(Float64(0.5 - t_0)) + 1.0));
	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], 1.0005], 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[(N[(N[(t$95$0 - 0.5), $MachinePrecision] - -1.0), $MachinePrecision] / N[(N[Sqrt[N[(0.5 - t$95$0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.00049999999999994

   1. Initial program 54.7%

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

   4. Applied rewrites 54.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 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 1.00049999999999994 < (hypot.f64 #s(literal 1 binary64) x)

   1. Initial program 98.3%

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right) - -1}{\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.9%

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 98.7

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

   7. Applied rewrites 98.7%

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

Alternative 8: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.9%

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 98.7

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

   7. Applied rewrites 98.7%

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

Alternative 9: 98.6% 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 54.9%

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

   4. Applied rewrites 54.9%

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

   5. 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 99.5

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

   7. Applied rewrites 99.5%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 98.7

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

   7. Applied rewrites 98.7%

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

Alternative 10: 97.9% accurate, 1.0× speedup

Math

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

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 54.9%

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

   4. Applied rewrites 54.9%

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

   5. 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 99.5

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

   7. Applied rewrites 99.5%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.2%

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

Alternative 11: 97.6% 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 54.9%

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

   4. Applied rewrites 54.9%

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

   5. 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 98.6

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

   7. Applied rewrites 98.6%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.2%

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

Alternative 12: 51.8% 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 73.6%

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

4. Applied rewrites 74.3%

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

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 57.9

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

7. Applied rewrites 57.9%

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

Reproduce

herbie shell --seed 2024331 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))