Given's Rotation SVD example, simplified

Percentage Accurate: 76.0% → 99.5%

Time: 9.7s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.0% 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: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.5%

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 1 < (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%

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

   6. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.9

         \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5, 1.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}, 1\right)}{1 - {\left(\frac{0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5\right)}^{3}} \cdot \left(\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} + 1\right)}} \]

   10. Applied rewrites 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	t_1 = Float64(t_0 + 0.5)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = Float64(Float64(1.0 - (t_1 ^ 3.0)) / Float64(Float64(sqrt(t_1) + 1.0) * fma(t_1, Float64(1.5 + 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]}, Block[{t$95$1 = N[(t$95$0 + 0.5), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[t$95$1], $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 * N[(1.5 + t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.5%

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 1 < (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. 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. associate--l- N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25}{0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)} - \frac{{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}^{2}}{0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   8. Applied rewrites 99.9%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.5%

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 1 < (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%

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

   6. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-hypot.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. lift-hypot.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.9%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0) {
		tmp = 0.125 * (x * x);
	} else {
		tmp = (0.5 - t_0) / (Math.sqrt((t_0 + 0.5)) + 1.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0:
		tmp = 0.125 * (x * x)
	else:
		tmp = (0.5 - t_0) / (math.sqrt((t_0 + 0.5)) + 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) <= 1.0)
		tmp = Float64(0.125 * Float64(x * x));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(sqrt(Float64(t_0 + 0.5)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0)
		tmp = 0.125 * (x * x);
	else
		tmp = (0.5 - t_0) / (sqrt((t_0 + 0.5)) + 1.0);
	end
	tmp_2 = 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], 1.0], N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(N[Sqrt[N[(t$95$0 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.5%

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

   4. Applied rewrites 64.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 1 < (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. 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. associate--l- N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25}{0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)} - \frac{{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}^{2}}{0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   8. Applied rewrites 99.8%

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

Alternative 5: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{x} + 0.5\\ \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x) 0.5)))
   (if (<= (pow (hypot 1.0 x) -1.0) 0.05)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0))
     (*
      (fma
       (fma (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x -0.0859375)
       (* x x)
       0.125)
      (* x x)))))

C

double code(double x) {
	double t_0 = (0.5 / x) + 0.5;
	double tmp;
	if (pow(hypot(1.0, x), -1.0) <= 0.05) {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	} else {
		tmp = fma(fma((fma(-0.056243896484375, (x * x), 0.0673828125) * x), x, -0.0859375), (x * x), 0.125) * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(0.5 / x) + 0.5)
	tmp = 0.0
	if ((hypot(1.0, x) ^ -1.0) <= 0.05)
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	else
		tmp = Float64(fma(fma(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x), x, -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.5 / x), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -1.0], $MachinePrecision], 0.05], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 0.050000000000000003

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 98.7%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\frac{1 - \left(\frac{0.5}{x} + 0.5\right)}{\sqrt{\frac{0.5}{x} + 0.5} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.25}{x}, \sqrt{2}, \frac{0.5}{\sqrt{0.5} + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (pow (hypot 1.0 x) -1.0) 0.05)
   (fma (/ -0.25 x) (sqrt 2.0) (/ 0.5 (+ (sqrt 0.5) 1.0)))
   (*
    (fma
     (fma (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x -0.0859375)
     (* x x)
     0.125)
    (* x x))))

C

double code(double x) {
	double tmp;
	if (pow(hypot(1.0, x), -1.0) <= 0.05) {
		tmp = fma((-0.25 / x), sqrt(2.0), (0.5 / (sqrt(0.5) + 1.0)));
	} else {
		tmp = fma(fma((fma(-0.056243896484375, (x * x), 0.0673828125) * x), x, -0.0859375), (x * x), 0.125) * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((hypot(1.0, x) ^ -1.0) <= 0.05)
		tmp = fma(Float64(-0.25 / x), sqrt(2.0), Float64(0.5 / Float64(sqrt(0.5) + 1.0)));
	else
		tmp = Float64(fma(fma(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x), x, -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -1.0], $MachinePrecision], 0.05], N[(N[(-0.25 / x), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 0.050000000000000003

   1. Initial program 98.4%

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

   7. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.25}{x}, \sqrt{2}, 1 - \sqrt{0.5}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.7%

      \[\leadsto \mathsf{fma}\left(\frac{-0.25}{x}, \sqrt{2}, \frac{0.5}{\sqrt{0.5} + 1}\right) \]

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.25}{x}, \sqrt{2}, \frac{0.5}{\sqrt{0.5} + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (pow (hypot 1.0 x) -1.0) 0.05)
   (/ 0.5 (+ (sqrt 0.5) 1.0))
   (*
    (fma
     (fma (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x -0.0859375)
     (* x x)
     0.125)
    (* x x))))

C

double code(double x) {
	double tmp;
	if (pow(hypot(1.0, x), -1.0) <= 0.05) {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	} else {
		tmp = fma(fma((fma(-0.056243896484375, (x * x), 0.0673828125) * x), x, -0.0859375), (x * x), 0.125) * (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((hypot(1.0, x) ^ -1.0) <= 0.05)
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	else
		tmp = Float64(fma(fma(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x), x, -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -1.0], $MachinePrecision], 0.05], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 0.050000000000000003

   1. Initial program 98.4%

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

   4. Applied rewrites 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 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x, x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (pow (hypot 1.0 x) -1.0) 0.05)
   (/ 0.5 (+ (sqrt 0.5) 1.0))
   (* (* (fma (fma 0.0673828125 (* x x) -0.0859375) (* x x) 0.125) x) x)))

C

double code(double x) {
	double tmp;
	if (pow(hypot(1.0, x), -1.0) <= 0.05) {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	} else {
		tmp = (fma(fma(0.0673828125, (x * x), -0.0859375), (x * x), 0.125) * x) * x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((hypot(1.0, x) ^ -1.0) <= 0.05)
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	else
		tmp = Float64(Float64(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * x) * x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], -1.0], $MachinePrecision], 0.05], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.0673828125 * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)) < 0.050000000000000003

   1. Initial program 98.4%

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

   4. Applied rewrites 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 98.4

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

   7. Applied rewrites 98.4%

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

   if 0.050000000000000003 < (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))

   1. Initial program 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-1} \leq 0.05:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0673828125, x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 0.8× speedup

Math

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

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))
   (fma (/ (sqrt 0.5) x) (+ -0.5 (/ 0.125 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 = fma((sqrt(0.5) / x), (-0.5 + (0.125 / x)), (0.5 / (sqrt(0.5) + 1.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(fma(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x), x, -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = fma(Float64(sqrt(0.5) / x), Float64(-0.5 + Float64(0.125 / x)), 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] * x), $MachinePrecision] * x + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[0.5], $MachinePrecision] / x), $MachinePrecision] * N[(-0.5 + N[(0.125 / x), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.2%

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

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

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

      1. associate--r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate--l+ N/A

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

      7. associate-+r+ N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 97.2%

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

   7. Applied rewrites 98.7%

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

Alternative 10: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (fma (* (* x x) -0.0859375) 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(((x * x) * -0.0859375), x, (0.125 * x)) * x;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(fma(Float64(Float64(x * x) * -0.0859375), x, Float64(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[(x * x), $MachinePrecision] * -0.0859375), $MachinePrecision] * x + N[(0.125 * x), $MachinePrecision]), $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 64.9%

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

   4. Applied rewrites 64.9%

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

      1. lift--.f64 N/A

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

      2. 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. associate--l- N/A

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

      4. flip-- N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. div-sub N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 64.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25}{0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)} - \frac{{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}^{2}}{0.5 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 1\right)}}}{\left(-\sqrt{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right) - 1} \]

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 98.9%

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

   11. Applied rewrites 98.9%

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

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

   1. Initial program 98.4%

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

   4. Applied rewrites 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 98.4

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

   7. Applied rewrites 98.4%

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

Alternative 11: 98.7% accurate, 1.0× speedup

Math

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

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 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

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

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 98.4

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

   7. Applied rewrites 98.4%

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

Alternative 12: 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 64.9%

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.9%

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

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

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 96.9%

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

Alternative 13: 97.7% 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 64.9%

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

   4. Applied rewrites 64.9%

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

   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 98.8

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

   7. Applied rewrites 98.8%

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

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

   1. Initial program 98.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.9%

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

Alternative 14: 51.4% 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 82.6%

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

4. Applied rewrites 83.4%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot {x}^{2}} \]
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 48.9

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

7. Applied rewrites 48.9%

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

Alternative 15: 27.4% accurate, 33.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.6%

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

4. Applied rewrites 82.6%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 51.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 31.6%

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

Reproduce

herbie shell --seed 2024321 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))