Given's Rotation SVD example, simplified

Percentage Accurate: 75.0% → 99.8%

Time: 8.7s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.0% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, 0.125 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + 0.5}{\sqrt{0.5 - \frac{-0.5}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (fma
    (pow x 4.0)
    (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
    (* 0.125 (* x x)))
   (/
    (+ (/ -0.5 (hypot 1.0 x)) 0.5)
    (+ (sqrt (- 0.5 (/ -0.5 (sqrt (fma x x 1.0))))) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma(pow(x, 4.0), (((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (0.125 * (x * x)));
	} else {
		tmp = ((-0.5 / hypot(1.0, x)) + 0.5) / (sqrt((0.5 - (-0.5 / sqrt(fma(x, x, 1.0))))) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = fma((x ^ 4.0), Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(0.125 * Float64(x * x)));
	else
		tmp = Float64(Float64(Float64(-0.5 / hypot(1.0, x)) + 0.5) / Float64(sqrt(Float64(0.5 - Float64(-0.5 / sqrt(fma(x, x, 1.0))))) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] + N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / N[(N[Sqrt[N[(0.5 - N[(-0.5 / N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, 0.125 \cdot \left(x \cdot x\right)\right)} \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

   8. Applied rewrites 100.0%

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

      1. lift-hypot.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-hypot.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lift-hypot.f64 N/A

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

      7. lift-hypot.f64 N/A

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

      8. rem-square-sqrt N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 100.0

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

   10. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{\frac{0.25}{x \cdot x} - 0.5}{x}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, 0.125 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ (- (/ 0.25 (* x x)) 0.5) x))))
   (if (<= (hypot 1.0 x) 2.0)
     (fma
      (pow x 4.0)
      (- (* (* (fma -0.056243896484375 (* x x) 0.0673828125) x) x) 0.0859375)
      (* 0.125 (* x x)))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

double code(double x) {
	double t_0 = 0.5 - (((0.25 / (x * x)) - 0.5) / x);
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = fma(pow(x, 4.0), (((fma(-0.056243896484375, (x * x), 0.0673828125) * x) * x) - 0.0859375), (0.125 * (x * x)));
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(Float64(Float64(0.25 / Float64(x * x)) - 0.5) / x))
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = fma((x ^ 4.0), Float64(Float64(Float64(fma(-0.056243896484375, Float64(x * x), 0.0673828125) * x) * x) - 0.0859375), Float64(0.125 * Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(N[(N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(N[(-0.056243896484375 * N[(x * x), $MachinePrecision] + 0.0673828125), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.0859375), $MachinePrecision] + N[(0.125 * N[(x * x), $MachinePrecision]), $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 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, \left(\mathsf{fma}\left(-0.056243896484375, x \cdot x, 0.0673828125\right) \cdot x\right) \cdot x - 0.0859375, 0.125 \cdot \left(x \cdot x\right)\right)} \]

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

      \[\leadsto 1 - \sqrt{0.5 - \color{blue}{\frac{\frac{0.25}{x \cdot x} - 0.5}{x}}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 98.6%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - \frac{\frac{0.25}{x \cdot x} - 0.5}{x}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(0.0673828125 \cdot \left(x \cdot x\right) - 0.0859375, x \cdot x, 0.125\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 0.5 (/ (- (/ 0.25 (* x x)) 0.5) x))))
   (if (<= (hypot 1.0 x) 2.0)
     (* (* (fma (- (* 0.0673828125 (* x x)) 0.0859375) (* x x) 0.125) x) x)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

double code(double x) {
	double t_0 = 0.5 - (((0.25 / (x * x)) - 0.5) / x);
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(((0.0673828125 * (x * x)) - 0.0859375), (x * x), 0.125) * x) * x;
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(0.5 - Float64(Float64(Float64(0.25 / Float64(x * x)) - 0.5) / x))
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(Float64(Float64(0.0673828125 * Float64(x * x)) - 0.0859375), Float64(x * x), 0.125) * x) * x);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 - N[(N[(N[(0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(0.0673828125 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.0859375), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] * x), $MachinePrecision] * x), $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 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

   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. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

      \[\leadsto 1 - \sqrt{0.5 - \color{blue}{\frac{\frac{0.25}{x \cdot x} - 0.5}{x}}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 98.6%

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

Alternative 4: 98.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

   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. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 97.2

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

   7. Applied rewrites 97.2%

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

Alternative 5: 98.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

   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. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   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

      \[\leadsto 1 - \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{x}}} \]
   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 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 6: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   8. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

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

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 7: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (fma (* -0.0859375 (* x 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)), 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(-0.0859375 * Float64(x * x)), 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[(-0.0859375 * N[(x * x), $MachinePrecision]), $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 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 45.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{8} + \frac{-11}{128} \cdot {x}^{2}\right)} \]
   8. 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. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   11. Applied rewrites 99.9%

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

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

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 97.5

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

   11. Applied rewrites 97.5%

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

Alternative 8: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. sqrt-pow2 N/A

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

      7. lift--.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-hypot.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 97.5

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

   11. Applied rewrites 97.5%

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

Alternative 9: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

   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 99.9

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 96.1%

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

Alternative 10: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (x * x) * 0.125;
	} 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 = (x * x) * 0.125;
	} 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 = (x * x) * 0.125
	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(Float64(x * x) * 0.125);
	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 = (x * x) * 0.125;
	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[(N[(x * x), $MachinePrecision] * 0.125), $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 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower--.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. metadata-eval 45.6

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   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 96.1%

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

Alternative 11: 51.7% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 69.4

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 56.6

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

7. Applied rewrites 56.6%

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

Alternative 12: 26.8% 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 69.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. distribute-lft-in N/A

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

   4. metadata-eval N/A

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

   5. fp-cancel-sign-sub-inv N/A

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

   6. lower--.f64 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 69.4

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

4. Applied rewrites 69.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 25.9%

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

Reproduce

herbie shell --seed 2024344 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))