Given's Rotation SVD example, simplified

Percentage Accurate: 76.6% → 99.4%

Time: 8.3s

Alternatives: 9

Speedup: 6.7×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ t_1 := \sqrt{t\_0}\\ t_2 := t\_1 + 1\\ t_3 := \mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\\ t_4 := {t\_3}^{2} \cdot \sqrt{2}\\ t_5 := \frac{\sqrt{0.5}}{t\_4}\\ t_6 := \frac{0.1875}{t\_3}\\ \mathbf{if}\;x\_m \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.15625}{t\_3} - \mathsf{fma}\left(\frac{0.25}{\sqrt{2}}, \frac{\mathsf{fma}\left(t\_5, -0.0625, t\_6\right) \cdot \sqrt{0.5}}{t\_3}, \frac{0.125 \cdot \left(0.34375 \cdot \sqrt{0.5}\right)}{t\_4}\right), x\_m \cdot x\_m, 0.0625 \cdot t\_5\right) - t\_6, x\_m \cdot x\_m, \frac{0.25}{t\_3}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - \mathsf{fma}\left(t\_1, t\_0, t\_0\right)}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5))
        (t_1 (sqrt t_0))
        (t_2 (+ t_1 1.0))
        (t_3 (fma (sqrt 2.0) (sqrt 0.5) 1.0))
        (t_4 (* (pow t_3 2.0) (sqrt 2.0)))
        (t_5 (/ (sqrt 0.5) t_4))
        (t_6 (/ 0.1875 t_3)))
   (if (<= x_m 1.2)
     (*
      (fma
       (-
        (fma
         (-
          (/ 0.15625 t_3)
          (fma
           (/ 0.25 (sqrt 2.0))
           (/ (* (fma t_5 -0.0625 t_6) (sqrt 0.5)) t_3)
           (/ (* 0.125 (* 0.34375 (sqrt 0.5))) t_4)))
         (* x_m x_m)
         (* 0.0625 t_5))
        t_6)
       (* x_m x_m)
       (/ 0.25 t_3))
      (* x_m x_m))
     (/ (- t_2 (fma t_1 t_0 t_0)) (* t_2 t_2)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double t_1 = sqrt(t_0);
	double t_2 = t_1 + 1.0;
	double t_3 = fma(sqrt(2.0), sqrt(0.5), 1.0);
	double t_4 = pow(t_3, 2.0) * sqrt(2.0);
	double t_5 = sqrt(0.5) / t_4;
	double t_6 = 0.1875 / t_3;
	double tmp;
	if (x_m <= 1.2) {
		tmp = fma((fma(((0.15625 / t_3) - fma((0.25 / sqrt(2.0)), ((fma(t_5, -0.0625, t_6) * sqrt(0.5)) / t_3), ((0.125 * (0.34375 * sqrt(0.5))) / t_4))), (x_m * x_m), (0.0625 * t_5)) - t_6), (x_m * x_m), (0.25 / t_3)) * (x_m * x_m);
	} else {
		tmp = (t_2 - fma(t_1, t_0, t_0)) / (t_2 * t_2);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	t_1 = sqrt(t_0)
	t_2 = Float64(t_1 + 1.0)
	t_3 = fma(sqrt(2.0), sqrt(0.5), 1.0)
	t_4 = Float64((t_3 ^ 2.0) * sqrt(2.0))
	t_5 = Float64(sqrt(0.5) / t_4)
	t_6 = Float64(0.1875 / t_3)
	tmp = 0.0
	if (x_m <= 1.2)
		tmp = Float64(fma(Float64(fma(Float64(Float64(0.15625 / t_3) - fma(Float64(0.25 / sqrt(2.0)), Float64(Float64(fma(t_5, -0.0625, t_6) * sqrt(0.5)) / t_3), Float64(Float64(0.125 * Float64(0.34375 * sqrt(0.5))) / t_4))), Float64(x_m * x_m), Float64(0.0625 * t_5)) - t_6), Float64(x_m * x_m), Float64(0.25 / t_3)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(t_2 - fma(t_1, t_0, t_0)) / Float64(t_2 * t_2));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[t$95$3, 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[0.5], $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(0.1875 / t$95$3), $MachinePrecision]}, If[LessEqual[x$95$m, 1.2], N[(N[(N[(N[(N[(N[(0.15625 / t$95$3), $MachinePrecision] - N[(N[(0.25 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$5 * -0.0625 + t$95$6), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(N[(0.125 * N[(0.34375 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.0625 * t$95$5), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.25 / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - N[(t$95$1 * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.19999999999999996

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{5}{32} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} - \left(\frac{1}{8} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{3}{8} - \frac{1}{16} \cdot \frac{1}{{\left(\sqrt{2}\right)}^{2}}\right)}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{1}{4} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)}{\sqrt{2} \cdot \left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}\right)\right) - \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 71.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.15625}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)} - \mathsf{fma}\left(\frac{0.25}{\sqrt{2}}, \frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}, -0.0625, \frac{0.1875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right) \cdot \sqrt{0.5}}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}, \frac{0.125 \cdot \left(0.34375 \cdot \sqrt{0.5}\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}\right), x \cdot x, 0.0625 \cdot \frac{\sqrt{0.5}}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}\right) - \frac{0.1875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}, x \cdot x, \frac{0.25}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right) \cdot \left(x \cdot x\right)} \]

   if 1.19999999999999996 < 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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

       1. lift-*.f64 N/A

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

       2. *-lft-identity 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ t_1 := \sqrt{t\_0}\\ t_2 := t\_1 + 1\\ t_3 := \mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\\ \mathbf{if}\;x\_m \leq 1.12:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(\frac{\sqrt{0.5}}{{t\_3}^{2} \cdot \sqrt{2}}, -0.0625, \frac{0.1875}{t\_3}\right), \frac{0.25}{t\_3}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - \mathsf{fma}\left(t\_1, t\_0, t\_0\right)}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5))
        (t_1 (sqrt t_0))
        (t_2 (+ t_1 1.0))
        (t_3 (fma (sqrt 2.0) (sqrt 0.5) 1.0)))
   (if (<= x_m 1.12)
     (*
      (fma
       (* (- x_m) x_m)
       (fma (/ (sqrt 0.5) (* (pow t_3 2.0) (sqrt 2.0))) -0.0625 (/ 0.1875 t_3))
       (/ 0.25 t_3))
      (* x_m x_m))
     (/ (- t_2 (fma t_1 t_0 t_0)) (* t_2 t_2)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double t_1 = sqrt(t_0);
	double t_2 = t_1 + 1.0;
	double t_3 = fma(sqrt(2.0), sqrt(0.5), 1.0);
	double tmp;
	if (x_m <= 1.12) {
		tmp = fma((-x_m * x_m), fma((sqrt(0.5) / (pow(t_3, 2.0) * sqrt(2.0))), -0.0625, (0.1875 / t_3)), (0.25 / t_3)) * (x_m * x_m);
	} else {
		tmp = (t_2 - fma(t_1, t_0, t_0)) / (t_2 * t_2);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	t_1 = sqrt(t_0)
	t_2 = Float64(t_1 + 1.0)
	t_3 = fma(sqrt(2.0), sqrt(0.5), 1.0)
	tmp = 0.0
	if (x_m <= 1.12)
		tmp = Float64(fma(Float64(Float64(-x_m) * x_m), fma(Float64(sqrt(0.5) / Float64((t_3 ^ 2.0) * sqrt(2.0))), -0.0625, Float64(0.1875 / t_3)), Float64(0.25 / t_3)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(t_2 - fma(t_1, t_0, t_0)) / Float64(t_2 * t_2));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.12], N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Power[t$95$3, 2.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.0625 + N[(0.1875 / t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.25 / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 - N[(t$95$1 * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1200000000000001

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left({x}^{2} \cdot \left(\frac{-1}{16} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2} \cdot {\left(1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}} + \frac{3}{16} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)\right) + \frac{1}{4} \cdot \frac{1}{1 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right) \cdot {x}^{2}} \]

   10. Applied rewrites 70.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\frac{\sqrt{0.5}}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)\right)}^{2} \cdot \sqrt{2}}, -0.0625, \frac{0.1875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right), \frac{0.25}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 1\right)}\right) \cdot \left(x \cdot x\right)} \]

   if 1.1200000000000001 < 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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

       1. lift-*.f64 N/A

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

       2. *-lft-identity 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 3: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ t_1 := \sqrt{t\_0}\\ t_2 := t\_1 + 1\\ \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - \mathsf{fma}\left(t\_1, t\_0, t\_0\right)}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)) (t_1 (sqrt t_0)) (t_2 (+ t_1 1.0)))
   (if (<= x_m 1.25)
     (* (* x_m x_m) 0.125)
     (/ (- t_2 (fma t_1 t_0 t_0)) (* t_2 t_2)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double t_1 = sqrt(t_0);
	double t_2 = t_1 + 1.0;
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (t_2 - fma(t_1, t_0, t_0)) / (t_2 * t_2);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	t_1 = sqrt(t_0)
	t_2 = Float64(t_1 + 1.0)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(t_2 - fma(t_1, t_0, t_0)) / Float64(t_2 * t_2));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.25], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(t$95$2 - N[(t$95$1 * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. unpow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-sqrt.f64 70.9

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

   10. Applied rewrites 70.9%

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

   12. Applied rewrites 70.9%

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

   if 1.25 < 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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

       1. lift-*.f64 N/A

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

       2. *-lft-identity 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ t_1 := \sqrt{t\_0} + 1\\ \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot \left(1 - t\_0\right)}{t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)) (t_1 (+ (sqrt t_0) 1.0)))
   (if (<= x_m 1.25)
     (* (* x_m x_m) 0.125)
     (/ (* t_1 (- 1.0 t_0)) (* t_1 t_1)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double t_1 = sqrt(t_0) + 1.0;
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (t_1 * (1.0 - t_0)) / (t_1 * t_1);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (0.5d0 / x_m) + 0.5d0
    t_1 = sqrt(t_0) + 1.0d0
    if (x_m <= 1.25d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = (t_1 * (1.0d0 - t_0)) / (t_1 * t_1)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double t_1 = Math.sqrt(t_0) + 1.0;
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (t_1 * (1.0 - t_0)) / (t_1 * t_1);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (0.5 / x_m) + 0.5
	t_1 = math.sqrt(t_0) + 1.0
	tmp = 0
	if x_m <= 1.25:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = (t_1 * (1.0 - t_0)) / (t_1 * t_1)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	t_1 = Float64(sqrt(t_0) + 1.0)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(t_1 * Float64(1.0 - t_0)) / Float64(t_1 * t_1));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (0.5 / x_m) + 0.5;
	t_1 = sqrt(t_0) + 1.0;
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = (t_1 * (1.0 - t_0)) / (t_1 * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 1.25], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(N[(t$95$1 * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. unpow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-sqrt.f64 70.9

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

   10. Applied rewrites 70.9%

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

   12. Applied rewrites 70.9%

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

   if 1.25 < 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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

Alternative 5: 99.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5}{x\_m} + 0.5\\ \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ (/ 0.5 x_m) 0.5)))
   (if (<= x_m 1.25)
     (* (* x_m x_m) 0.125)
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 / x_m) + 0.5d0
    if (x_m <= 1.25d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = (1.0d0 - t_0) / (sqrt(t_0) + 1.0d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = (0.5 / x_m) + 0.5;
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = (1.0 - t_0) / (Math.sqrt(t_0) + 1.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = (0.5 / x_m) + 0.5
	tmp = 0
	if x_m <= 1.25:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = (1.0 - t_0) / (math.sqrt(t_0) + 1.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(0.5 / x_m) + 0.5)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = (0.5 / x_m) + 0.5;
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.25], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $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 x < 1.25

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. unpow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-sqrt.f64 70.9

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

   10. Applied rewrites 70.9%

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

   12. Applied rewrites 70.9%

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

   if 1.25 < 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 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 6: 98.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.25:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} + 0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.25) (* (* x_m x_m) 0.125) (- 1.0 (sqrt (+ (/ 0.5 x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.25d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(((0.5d0 / x_m) + 0.5d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.25) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(((0.5 / x_m) + 0.5));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.25:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt(((0.5 / x_m) + 0.5))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.25)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) + 0.5)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.25)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt(((0.5 / x_m) + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.25], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(0.5 / x$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. unpow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-sqrt.f64 70.9

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

   10. Applied rewrites 70.9%

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

   12. Applied rewrites 70.9%

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

   if 1.25 < 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 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 7: 98.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* (* x_m x_m) 0.125) (/ 0.5 (+ (sqrt 0.5) 1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 0.5d0 / (sqrt(0.5d0) + 1.0d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 0.5 / (Math.sqrt(0.5) + 1.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 0.5 / (math.sqrt(0.5) + 1.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(0.5 / Float64(sqrt(0.5) + 1.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 0.5 / (sqrt(0.5) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. unpow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-sqrt.f64 70.9

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

   10. Applied rewrites 70.9%

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

   12. Applied rewrites 70.9%

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

   if 1.55000000000000004 < 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 96.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.0%

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

Alternative 8: 97.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55:\\ \;\;\;\;\left(x\_m \cdot x\_m\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.55) (* (* x_m x_m) 0.125) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.55d0) then
        tmp = (x_m * x_m) * 0.125d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		tmp = (x_m * x_m) * 0.125;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = (x_m * x_m) * 0.125
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		tmp = Float64(Float64(x_m * x_m) * 0.125);
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.55)
		tmp = (x_m * x_m) * 0.125;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.55], N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 63.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 32.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. rem-square-sqrt N/A

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

      11. unpow2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. unpow2 N/A

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

      14. rem-square-sqrt N/A

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

      15. lower-sqrt.f64 70.9

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

   10. Applied rewrites 70.9%

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

   12. Applied rewrites 70.9%

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

   if 1.55000000000000004 < 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 96.6%

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

Alternative 9: 50.9% accurate, 12.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (* x_m x_m) 0.125))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * 0.125;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * 0.125;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * 0.125

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * 0.125)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.125), $MachinePrecision]

Derivation

1. Initial program 72.1%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 48.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

7. Applied rewrites 48.8%

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

8. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. rem-square-sqrt N/A

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

   11. unpow2 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. unpow2 N/A

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

   14. rem-square-sqrt N/A

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

   15. lower-sqrt.f64 54.6

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

10. Applied rewrites 54.6%

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

12. Applied rewrites 54.6%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))