Given's Rotation SVD example, simplified

Percentage Accurate: 73.8% → 99.7%

Time: 8.3s

Alternatives: 8

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: 73.8% 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: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{0.5} + 1\\ t_1 := {t\_0}^{2}\\ t_2 := \frac{0.5}{t\_0}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.17724609375, x \cdot x, 0.1953125\right), x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{t\_2}{{x}^{4}} + t\_2\right) - \mathsf{fma}\left(\frac{\frac{-0.25}{{x}^{4}}}{\sqrt{0.5}}, \frac{0.625}{t\_1} + \frac{\frac{\frac{0.125}{\sqrt{0.5} + 0.5}}{t\_0} + t\_2}{t\_0}, \frac{\frac{0.125}{t\_1}}{\left(\sqrt{0.5} \cdot x\right) \cdot x} + \frac{\frac{0.5}{\mathsf{fma}\left(\sqrt{0.5}, x, x\right)}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (sqrt 0.5) 1.0)) (t_1 (pow t_0 2.0)) (t_2 (/ 0.5 t_0)))
   (if (<= (hypot 1.0 x) 2.0)
     (*
      (*
       (fma
        (fma (fma -0.17724609375 (* x x) 0.1953125) (* x x) -0.21875)
        (* x x)
        0.25)
       x)
      x)
     (-
      (+ (/ t_2 (pow x 4.0)) t_2)
      (fma
       (/ (/ -0.25 (pow x 4.0)) (sqrt 0.5))
       (+ (/ 0.625 t_1) (/ (+ (/ (/ 0.125 (+ (sqrt 0.5) 0.5)) t_0) t_2) t_0))
       (+
        (/ (/ 0.125 t_1) (* (* (sqrt 0.5) x) x))
        (/ (/ 0.5 (fma (sqrt 0.5) x x)) x)))))))

C

double code(double x) {
	double t_0 = sqrt(0.5) + 1.0;
	double t_1 = pow(t_0, 2.0);
	double t_2 = 0.5 / t_0;
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.17724609375, (x * x), 0.1953125), (x * x), -0.21875), (x * x), 0.25) * x) * x;
	} else {
		tmp = ((t_2 / pow(x, 4.0)) + t_2) - fma(((-0.25 / pow(x, 4.0)) / sqrt(0.5)), ((0.625 / t_1) + ((((0.125 / (sqrt(0.5) + 0.5)) / t_0) + t_2) / t_0)), (((0.125 / t_1) / ((sqrt(0.5) * x) * x)) + ((0.5 / fma(sqrt(0.5), x, x)) / x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(sqrt(0.5) + 1.0)
	t_1 = t_0 ^ 2.0
	t_2 = Float64(0.5 / t_0)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.17724609375, Float64(x * x), 0.1953125), Float64(x * x), -0.21875), Float64(x * x), 0.25) * x) * x);
	else
		tmp = Float64(Float64(Float64(t_2 / (x ^ 4.0)) + t_2) - fma(Float64(Float64(-0.25 / (x ^ 4.0)) / sqrt(0.5)), Float64(Float64(0.625 / t_1) + Float64(Float64(Float64(Float64(0.125 / Float64(sqrt(0.5) + 0.5)) / t_0) + t_2) / t_0)), Float64(Float64(Float64(0.125 / t_1) / Float64(Float64(sqrt(0.5) * x) * x)) + Float64(Float64(0.5 / fma(sqrt(0.5), x, x)) / x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.17724609375 * N[(x * x), $MachinePrecision] + 0.1953125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.21875), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(t$95$2 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] - N[(N[(N[(-0.25 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(0.625 / t$95$1), $MachinePrecision] + N[(N[(N[(N[(0.125 / N[(N[Sqrt[0.5], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.125 / t$95$1), $MachinePrecision] / N[(N[(N[Sqrt[0.5], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] * x + x), $MachinePrecision]), $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 52.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.17724609375, x \cdot x, 0.1953125\right), x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

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

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{x}^{4} \cdot \left(1 + \sqrt{\frac{1}{2}}\right)} + \frac{1}{2} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}}}\right) - \left(\frac{-1}{4} \cdot \frac{\frac{1}{2} + \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{\frac{1}{2}}\right)}^{2}}}{{x}^{4} \cdot \left(\sqrt{\frac{1}{2}} \cdot {\left(1 + \sqrt{\frac{1}{2}}\right)}^{2}\right)} + \left(\frac{-1}{4} \cdot \frac{\frac{1}{8} \cdot \frac{1}{\sqrt{\frac{1}{2}} \cdot {\left(1 + \sqrt{\frac{1}{2}}\right)}^{2}} + \frac{1}{2} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}}}}{{x}^{4} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left(1 + \sqrt{\frac{1}{2}}\right)\right)} + \left(\frac{\frac{1}{8}}{{x}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot {\left(1 + \sqrt{\frac{1}{2}}\right)}^{2}\right)} + \frac{\frac{1}{2}}{{x}^{2} \cdot \left(1 + \sqrt{\frac{1}{2}}\right)}\right)\right)\right)} \]

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{0.5}{\sqrt{0.5} + 1}}{{x}^{4}} + \frac{0.5}{\sqrt{0.5} + 1}\right) - \mathsf{fma}\left(\frac{\frac{-0.25}{{x}^{4}}}{\sqrt{0.5}}, \frac{\frac{0.5}{\sqrt{0.5} + 1} + \frac{\frac{0.125}{\sqrt{0.5} + 0.5}}{\sqrt{0.5} + 1}}{\sqrt{0.5} + 1} + \frac{0.625}{{\left(\sqrt{0.5} + 1\right)}^{2}}, \frac{\frac{0.5}{\mathsf{fma}\left(\sqrt{0.5}, x, x\right)}}{x} + \frac{\frac{0.125}{{\left(\sqrt{0.5} + 1\right)}^{2}}}{\left(\sqrt{0.5} \cdot x\right) \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.17724609375, x \cdot x, 0.1953125\right), x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.5}{\sqrt{0.5} + 1}}{{x}^{4}} + \frac{0.5}{\sqrt{0.5} + 1}\right) - \mathsf{fma}\left(\frac{\frac{-0.25}{{x}^{4}}}{\sqrt{0.5}}, \frac{0.625}{{\left(\sqrt{0.5} + 1\right)}^{2}} + \frac{\frac{\frac{0.125}{\sqrt{0.5} + 0.5}}{\sqrt{0.5} + 1} + \frac{0.5}{\sqrt{0.5} + 1}}{\sqrt{0.5} + 1}, \frac{\frac{0.125}{{\left(\sqrt{0.5} + 1\right)}^{2}}}{\left(\sqrt{0.5} \cdot x\right) \cdot x} + \frac{\frac{0.5}{\mathsf{fma}\left(\sqrt{0.5}, x, x\right)}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.17724609375, x \cdot x, 0.1953125\right), x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (*
    (*
     (fma
      (fma (fma -0.17724609375 (* x x) 0.1953125) (* x x) -0.21875)
      (* x x)
      0.25)
     x)
    x)
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(fma(-0.17724609375, (x * x), 0.1953125), (x * x), -0.21875), (x * x), 0.25) * x) * x;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(fma(-0.17724609375, Float64(x * x), 0.1953125), Float64(x * x), -0.21875), Float64(x * x), 0.25) * x) * x);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(N[(-0.17724609375 * N[(x * x), $MachinePrecision] + 0.1953125), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.21875), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{25}{128} + \frac{-363}{2048} \cdot {x}^{2}\right) - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.17724609375, x \cdot x, 0.1953125\right), x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

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

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma (fma 0.1953125 (* x x) -0.21875) (* x x) 0.25) x) x)
   (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(fma(0.1953125, (x * x), -0.21875), (x * x), 0.25) * x) * x;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(fma(fma(0.1953125, Float64(x * x), -0.21875), Float64(x * x), 0.25) * x) * x);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(N[(N[(0.1953125 * N[(x * x), $MachinePrecision] + -0.21875), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right) + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      9. sub-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{25}{128} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{7}{32}\right)\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{25}{128} \cdot {x}^{2} + \color{blue}{\frac{-7}{32}}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{25}{128}, {x}^{2}, \frac{-7}{32}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, \color{blue}{x \cdot x}, \frac{-7}{32}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, \color{blue}{x \cdot x}, \frac{-7}{32}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, x \cdot x, \frac{-7}{32}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      15. lower-*.f64 99.7

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

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

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.21875, x \cdot x, 0.25\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.21875 (* x x) 0.25) 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(-0.21875, (x * x), 0.25) * 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(-0.21875, Float64(x * x), 0.25) * 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[(-0.21875 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-7}{32} \cdot {x}^{2} + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-7}{32}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-7}{32}, \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.6

         \[\leadsto \left(\mathsf{fma}\left(-0.21875, \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.21875, x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

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

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-sqrt.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.21875, x \cdot x, 0.25\right) \cdot x\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (* (* (fma -0.21875 (* x x) 0.25) x) x)
   (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (fma(-0.21875, (x * x), 0.25) * 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(-0.21875, Float64(x * x), 0.25) * 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[(-0.21875 * N[(x * x), $MachinePrecision] + 0.25), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + \frac{-7}{32} \cdot {x}^{2}\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{-7}{32} \cdot {x}^{2} + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{-7}{32}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      8. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{-7}{32}, \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      9. lower-*.f64 99.6

         \[\leadsto \left(\mathsf{fma}\left(-0.21875, \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-0.21875, x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

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

   1. Initial program 97.6%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 98.1%

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

Alternative 6: 98.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* (* 0.25 x) x) (- 1.0 (sqrt 0.5))))

C

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

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (0.25 * x) * x;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (0.25 * x) * x
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (0.25 * x) * x;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 52.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

      11. associate--r+ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 52.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot {x}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right)} \cdot x \]

      6. +-commutative N/A

         \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right) + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\color{blue}{\left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

      9. sub-neg N/A

         \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{25}{128} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{7}{32}\right)\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      10. metadata-eval N/A

          \[\leadsto \left(\mathsf{fma}\left(\frac{25}{128} \cdot {x}^{2} + \color{blue}{\frac{-7}{32}}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{25}{128}, {x}^{2}, \frac{-7}{32}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      12. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, \color{blue}{x \cdot x}, \frac{-7}{32}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, \color{blue}{x \cdot x}, \frac{-7}{32}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, x \cdot x, \frac{-7}{32}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

      15. lower-*.f64 99.7

          \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\frac{1}{4} \cdot x\right) \cdot x \]
   9. Step-by-step derivation

   10. Applied rewrites 99.0%

       \[\leadsto \left(0.25 \cdot x\right) \cdot x \]

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

   1. Initial program 97.6%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 98.1%

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

Alternative 7: 51.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \left(0.25 \cdot x\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 0.25 x) x))

C

double code(double x) {
	return (0.25 * x) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.25d0 * x) * x
end function

Java

public static double code(double x) {
	return (0.25 * x) * x;
}

Python

def code(x):
	return (0.25 * x) * x

Julia

function code(x)
	return Float64(Float64(0.25 * x) * x)
end

MATLAB

function tmp = code(x)
	tmp = (0.25 * x) * x;
end

Wolfram

code[x_] := N[(N[(0.25 * x), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

   11. associate--r+ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot {x}^{2}} \]

   2. unpow2 N/A

      \[\leadsto \left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right) \cdot x} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{4} + {x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right)\right) \cdot x\right)} \cdot x \]

   6. +-commutative N/A

      \[\leadsto \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right) + \frac{1}{4}\right)} \cdot x\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\color{blue}{\left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}\right) \cdot {x}^{2}} + \frac{1}{4}\right) \cdot x\right) \cdot x \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\frac{25}{128} \cdot {x}^{2} - \frac{7}{32}, {x}^{2}, \frac{1}{4}\right)} \cdot x\right) \cdot x \]

   9. sub-neg N/A

      \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\frac{25}{128} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{7}{32}\right)\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

   10. metadata-eval N/A

       \[\leadsto \left(\mathsf{fma}\left(\frac{25}{128} \cdot {x}^{2} + \color{blue}{\frac{-7}{32}}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{25}{128}, {x}^{2}, \frac{-7}{32}\right)}, {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

   12. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, \color{blue}{x \cdot x}, \frac{-7}{32}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

   13. lower-*.f64 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, \color{blue}{x \cdot x}, \frac{-7}{32}\right), {x}^{2}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

   14. unpow2 N/A

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{25}{128}, x \cdot x, \frac{-7}{32}\right), \color{blue}{x \cdot x}, \frac{1}{4}\right) \cdot x\right) \cdot x \]

   15. lower-*.f64 50.4

       \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), \color{blue}{x \cdot x}, 0.25\right) \cdot x\right) \cdot x \]

7. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.1953125, x \cdot x, -0.21875\right), x \cdot x, 0.25\right) \cdot x\right) \cdot x} \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{4} \cdot x\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 50.4%

    \[\leadsto \left(0.25 \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 8: 51.7% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ 0.25 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.25 (* x x)))

C

double code(double x) {
	return 0.25 * (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0 * (x * x)
end function

Java

public static double code(double x) {
	return 0.25 * (x * x);
}

Python

def code(x):
	return 0.25 * (x * x)

Julia

function code(x)
	return Float64(0.25 * Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = 0.25 * (x * x);
end

Wolfram

code[x_] := N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. rem-square-sqrt N/A

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

   11. associate--r+ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 76.5%

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

5. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

      \[\leadsto \frac{1}{4} \cdot \color{blue}{\left(x \cdot x\right)} \]

   3. lower-*.f64 50.4

      \[\leadsto 0.25 \cdot \color{blue}{\left(x \cdot x\right)} \]

7. Applied rewrites 50.4%

   \[\leadsto \color{blue}{0.25 \cdot \left(x \cdot x\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))