Given's Rotation SVD example, simplified

Percentage Accurate: 76.1% → 99.9%

Time: 9.3s

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: 76.1% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.125 - \frac{-0.125}{t\_0 \cdot \mathsf{fma}\left(x, x, 1\right)}, \frac{1}{\frac{-0.25}{t\_0} + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25\right)}, -1\right)}{-1 - \sqrt{0.5 - \frac{-0.5}{t\_0}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma x x 1.0))))
   (if (<= (hypot 1.0 x) 1.2)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/
      (fma
       (- 0.125 (/ -0.125 (* t_0 (fma x x 1.0))))
       (/ 1.0 (+ (/ -0.25 t_0) (+ (/ 0.25 (fma x x 1.0)) 0.25)))
       -1.0)
      (- -1.0 (sqrt (- 0.5 (/ -0.5 t_0))))))))

C

double code(double x) {
	double t_0 = sqrt(fma(x, x, 1.0));
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = fma((0.125 - (-0.125 / (t_0 * fma(x, x, 1.0)))), (1.0 / ((-0.25 / t_0) + ((0.25 / fma(x, x, 1.0)) + 0.25))), -1.0) / (-1.0 - sqrt((0.5 - (-0.5 / t_0))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(x, x, 1.0))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(fma(Float64(0.125 - Float64(-0.125 / Float64(t_0 * fma(x, x, 1.0)))), Float64(1.0 / Float64(Float64(-0.25 / t_0) + Float64(Float64(0.25 / fma(x, x, 1.0)) + 0.25))), -1.0) / Float64(-1.0 - sqrt(Float64(0.5 - Float64(-0.5 / t_0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.2], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.125 - N[(-0.125 / N[(t$95$0 * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(-0.25 / t$95$0), $MachinePrecision] + N[(N[(0.25 / N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.19999999999999996

   1. Initial program 52.3%

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.19999999999999996 < (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%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.125 - \frac{-0.125}{\mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}}, \frac{1}{\left(0.25 + \frac{0.25}{\mathsf{fma}\left(x, x, 1\right)}\right) + \frac{-0.25}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}, -1\right)}}{\left(-\sqrt{0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.125 - \frac{-0.125}{\sqrt{\mathsf{fma}\left(x, x, 1\right)} \cdot \mathsf{fma}\left(x, x, 1\right)}, \frac{1}{\frac{-0.25}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}} + \left(\frac{0.25}{\mathsf{fma}\left(x, x, 1\right)} + 0.25\right)}, -1\right)}{-1 - \sqrt{0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - 1}{-1 - \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ -0.5 (sqrt (fma x x 1.0))))))
   (if (<= (hypot 1.0 x) 1.2)
     (*
      (fma
       (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
       (* x x)
       0.125)
      (* x x))
     (/ (- t_0 1.0) (- -1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 - (-0.5 / sqrt(fma(x, x, 1.0)));
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = (t_0 - 1.0) / (-1.0 - sqrt(t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(-0.5 / sqrt(fma(x, x, 1.0))))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(Float64(t_0 - 1.0) / Float64(-1.0 - sqrt(t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(-0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.2], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(-1.0 - N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.19999999999999996

   1. Initial program 52.3%

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.19999999999999996 < (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%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\right) - 1}{-1 - \sqrt{0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.4%

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

   9. Applied rewrites 99.4%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. lft-mult-inverse N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 98.7%

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

Alternative 4: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right), x \cdot x, -0.0859375\right), x \cdot x, 0.125\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.2)
   (*
    (fma
     (fma (fma -0.056243896484375 (* x x) 0.0673828125) (* x x) -0.0859375)
     (* x x)
     0.125)
    (* x x))
   (- 1.0 (sqrt (- 0.5 (/ -0.5 (sqrt (fma x x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = fma(fma(fma(-0.056243896484375, (x * x), 0.0673828125), (x * x), -0.0859375), (x * x), 0.125) * (x * x);
	} else {
		tmp = 1.0 - sqrt((0.5 - (-0.5 / sqrt(fma(x, x, 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = Float64(fma(fma(fma(-0.056243896484375, Float64(x * x), 0.0673828125), Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 - Float64(-0.5 / sqrt(fma(x, x, 1.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.2], N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 #s(literal 1 binary64) x) < 1.19999999999999996

   1. Initial program 52.3%

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   if 1.19999999999999996 < (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 98.5%

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

Alternative 5: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.4%

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

   9. Applied rewrites 99.4%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. lft-mult-inverse N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 97.2%

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

Alternative 6: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.4%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. lft-mult-inverse N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 97.2%

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

Alternative 7: 98.5% accurate, 0.9× speedup

Math

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

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

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.4

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.4%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. lft-mult-inverse N/A

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

      10. lower--.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 97.2

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

   5. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-+.f64 N/A

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

   7. Applied rewrites 97.2%

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

Alternative 8: 98.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (fma (fma 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(fma(fma(0.0673828125, Float64(x * x), -0.0859375), Float64(x * x), 0.125) * Float64(x * x));
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.4

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

   7. Applied rewrites 99.4%

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

   9. Applied rewrites 99.4%

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

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

   1. Initial program 98.5%

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

   4. Applied rewrites 100.0%

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

   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 9: 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 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.4

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

   7. Applied rewrites 99.4%

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

   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%

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

   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 10: 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 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.2

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

   7. Applied rewrites 99.2%

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

   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%

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

   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: 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 52.6%

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

   4. Applied rewrites 52.6%

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

   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 99.2

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

   7. Applied rewrites 99.2%

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

   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 96.9%

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

Alternative 12: 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 52.6%

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

   4. Applied rewrites 52.6%

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

   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 98.9

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

   7. Applied rewrites 98.9%

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

   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 96.9%

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

Alternative 13: 51.5% accurate, 12.2× speedup

Math

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

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(Float64(0.125 * x) * x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

4. Applied rewrites 76.7%

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

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 49.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{8} \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 50.7%

    \[\leadsto \left(0.125 \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 14: 51.5% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

4. Applied rewrites 76.7%

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

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 50.7

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

7. Applied rewrites 50.7%

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

Alternative 15: 27.7% 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 75.9%

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

4. Applied rewrites 75.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 26.9%

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

Reproduce

herbie shell --seed 2024236 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))