Given's Rotation SVD example, simplified

Percentage Accurate: 75.6% → 99.9%

Time: 5.8s

Alternatives: 10

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.6% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\\ \mathbf{if}\;x\_m \leq 0.0022:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{t\_0}^{2}}, 0.375, \frac{0.3046875}{t\_0}\right), \frac{0.375}{t\_0}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x\_m, x\_m, 1\right)\right)}^{-0.5}, -0.5, 0.5\right)}{\sqrt{\mathsf{fma}\left(\cos \tan^{-1} x\_m, 0.5, 0.5\right)} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (sqrt 2.0) (sqrt 0.5) 2.0)))
   (if (<= x_m 0.0022)
     (*
      (fma
       (* (- x_m) x_m)
       (fma
        (/ (fma (/ (sqrt 0.5) (sqrt 2.0)) -0.25 -0.25) (pow t_0 2.0))
        0.375
        (/ 0.3046875 t_0))
       (/ 0.375 t_0))
      (* x_m x_m))
     (/
      (fma (pow (fma x_m x_m 1.0) -0.5) -0.5 0.5)
      (+ (sqrt (fma (cos (atan x_m)) 0.5 0.5)) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(sqrt(2.0), sqrt(0.5), 2.0);
	double tmp;
	if (x_m <= 0.0022) {
		tmp = fma((-x_m * x_m), fma((fma((sqrt(0.5) / sqrt(2.0)), -0.25, -0.25) / pow(t_0, 2.0)), 0.375, (0.3046875 / t_0)), (0.375 / t_0)) * (x_m * x_m);
	} else {
		tmp = fma(pow(fma(x_m, x_m, 1.0), -0.5), -0.5, 0.5) / (sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(sqrt(2.0), sqrt(0.5), 2.0)
	tmp = 0.0
	if (x_m <= 0.0022)
		tmp = Float64(fma(Float64(Float64(-x_m) * x_m), fma(Float64(fma(Float64(sqrt(0.5) / sqrt(2.0)), -0.25, -0.25) / (t_0 ^ 2.0)), 0.375, Float64(0.3046875 / t_0)), Float64(0.375 / t_0)) * Float64(x_m * x_m));
	else
		tmp = Float64(fma((fma(x_m, x_m, 1.0) ^ -0.5), -0.5, 0.5) / Float64(sqrt(fma(cos(atan(x_m)), 0.5, 0.5)) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.0022], N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.25 + -0.25), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * 0.375 + N[(0.3046875 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.375 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x$95$m * x$95$m + 1.0), $MachinePrecision], -0.5], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] / N[(N[Sqrt[N[(N[Cos[N[ArcTan[x$95$m], $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00220000000000000013

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}^{1.5}}{\left(1 + \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right) + \sqrt{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), \frac{0.375}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right) \cdot \left(x \cdot x\right)} \]

   if 0.00220000000000000013 < x

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      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. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 100.0%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\\ t_1 := \frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5\\ \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\left(-x\_m\right) \cdot x\_m, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{t\_0}^{2}}, 0.375, \frac{0.3046875}{t\_0}\right), \frac{0.375}{t\_0}\right) \cdot \left(x\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sqrt{t\_1} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (sqrt 2.0) (sqrt 0.5) 2.0))
        (t_1 (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))
   (if (<= x_m 1.0)
     (*
      (fma
       (* (- x_m) x_m)
       (fma
        (/ (fma (/ (sqrt 0.5) (sqrt 2.0)) -0.25 -0.25) (pow t_0 2.0))
        0.375
        (/ 0.3046875 t_0))
       (/ 0.375 t_0))
      (* x_m x_m))
     (/ (- 1.0 t_1) (+ (sqrt t_1) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(sqrt(2.0), sqrt(0.5), 2.0);
	double t_1 = ((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5;
	double tmp;
	if (x_m <= 1.0) {
		tmp = fma((-x_m * x_m), fma((fma((sqrt(0.5) / sqrt(2.0)), -0.25, -0.25) / pow(t_0, 2.0)), 0.375, (0.3046875 / t_0)), (0.375 / t_0)) * (x_m * x_m);
	} else {
		tmp = (1.0 - t_1) / (sqrt(t_1) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(sqrt(2.0), sqrt(0.5), 2.0)
	t_1 = Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = Float64(fma(Float64(Float64(-x_m) * x_m), fma(Float64(fma(Float64(sqrt(0.5) / sqrt(2.0)), -0.25, -0.25) / (t_0 ^ 2.0)), 0.375, Float64(0.3046875 / t_0)), Float64(0.375 / t_0)) * Float64(x_m * x_m));
	else
		tmp = Float64(Float64(1.0 - t_1) / Float64(sqrt(t_1) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.0], N[(N[(N[((-x$95$m) * x$95$m), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * -0.25 + -0.25), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * 0.375 + N[(0.3046875 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.375 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(N[Sqrt[t$95$1], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}^{1.5}}{\left(1 + \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right) + \sqrt{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \left({x}^{2} \cdot \left(\frac{39}{128} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}} + \frac{3}{8} \cdot \frac{\frac{-1}{4} \cdot \frac{\sqrt{\frac{1}{2}}}{\sqrt{2}} - \frac{1}{4}}{{\left(2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}\right)}^{2}}\right)\right) + \frac{3}{8} \cdot \frac{1}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}\right)} \]

   10. Applied rewrites 64.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left(-x\right) \cdot x, \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{\sqrt{0.5}}{\sqrt{2}}, -0.25, -0.25\right)}{{\left(\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)\right)}^{2}}, 0.375, \frac{0.3046875}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right), \frac{0.375}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\right) \cdot \left(x \cdot x\right)} \]

   if 1 < 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

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

   5. Applied rewrites 97.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 98.8%

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

Alternative 3: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(-0.25, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m)))))) 2e-16)
   (- 1.0 (sqrt (fma -0.25 (* x_m x_m) 1.0)))
   (- 1.0 (sqrt (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))))) <= 2e-16) {
		tmp = 1.0 - sqrt(fma(-0.25, (x_m * x_m), 1.0));
	} else {
		tmp = 1.0 - sqrt((((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m)))))) <= 2e-16)
		tmp = Float64(1.0 - sqrt(fma(-0.25, Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-16], N[(1.0 - N[Sqrt[N[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))) < 2e-16

   1. Initial program 54.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.6%

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

   if 2e-16 < (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))))

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.3%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(-0.25, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5 - \frac{0.25}{x \cdot x}}{x} - -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\_m\right)}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(-0.25, x\_m \cdot x\_m, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x\_m} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x_m)))))) 2e-16)
   (- 1.0 (sqrt (fma -0.25 (* x_m x_m) 1.0)))
   (- 1.0 (sqrt (- (/ 0.5 x_m) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if ((1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x_m)))))) <= 2e-16) {
		tmp = 1.0 - sqrt(fma(-0.25, (x_m * x_m), 1.0));
	} else {
		tmp = 1.0 - sqrt(((0.5 / x_m) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x_m)))))) <= 2e-16)
		tmp = Float64(1.0 - sqrt(fma(-0.25, Float64(x_m * x_m), 1.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(0.5 / x_m) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-16], N[(1.0 - N[Sqrt[N[(-0.25 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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 (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x)))))) < 2e-16

   1. Initial program 54.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 54.6%

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

   if 2e-16 < (-.f64 #s(literal 1 binary64) (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) (hypot.f64 #s(literal 1 binary64) x))))))

   1. Initial program 98.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 96.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \leq 2 \cdot 10^{-16}:\\ \;\;\;\;1 - \sqrt{\mathsf{fma}\left(-0.25, x \cdot x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5}{x} - -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5\\ \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\frac{0.375 \cdot \left(x\_m \cdot x\_m\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sqrt{t\_0} + 1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))
   (if (<= x_m 1.05)
     (/ (* 0.375 (* x_m x_m)) (fma (sqrt 2.0) (sqrt 0.5) 2.0))
     (/ (- 1.0 t_0) (+ (sqrt t_0) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = ((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5;
	double tmp;
	if (x_m <= 1.05) {
		tmp = (0.375 * (x_m * x_m)) / fma(sqrt(2.0), sqrt(0.5), 2.0);
	} else {
		tmp = (1.0 - t_0) / (sqrt(t_0) + 1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(Float64(0.375 * Float64(x_m * x_m)) / fma(sqrt(2.0), sqrt(0.5), 2.0));
	else
		tmp = Float64(Float64(1.0 - t_0) / Float64(sqrt(t_0) + 1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[x$95$m, 1.05], N[(N[(0.375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(N[Sqrt[t$95$0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}^{1.5}}{\left(1 + \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right) + \sqrt{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{8} \cdot \frac{{x}^{2}}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]

   if 1.05000000000000004 < 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

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

   5. Applied rewrites 97.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

   7. Applied rewrites 98.8%

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

Alternative 6: 98.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000102:\\ \;\;\;\;\frac{0.375 \cdot \left(x\_m \cdot x\_m\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\sqrt{\mathsf{fma}\left(x\_m, x\_m, 1\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000102)
   (/ (* 0.375 (* x_m x_m)) (fma (sqrt 2.0) (sqrt 0.5) 2.0))
   (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (sqrt (fma x_m x_m 1.0)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000102) {
		tmp = (0.375 * (x_m * x_m)) / fma(sqrt(2.0), sqrt(0.5), 2.0);
	} else {
		tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / sqrt(fma(x_m, x_m, 1.0))))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000102)
		tmp = Float64(Float64(0.375 * Float64(x_m * x_m)) / fma(sqrt(2.0), sqrt(0.5), 2.0));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / sqrt(fma(x_m, x_m, 1.0)))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000102], N[(N[(0.375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[N[(x$95$m * x$95$m + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.01999999999999999e-4

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}^{1.5}}{\left(1 + \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right) + \sqrt{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{8} \cdot \frac{{x}^{2}}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]

   if 1.01999999999999999e-4 < x

   1. Initial program 98.5%

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

      1. lift-hypot.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 7: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05:\\ \;\;\;\;\frac{0.375 \cdot \left(x\_m \cdot x\_m\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5 - \frac{0.25}{x\_m \cdot x\_m}}{x\_m} - -0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.05)
   (/ (* 0.375 (* x_m x_m)) (fma (sqrt 2.0) (sqrt 0.5) 2.0))
   (- 1.0 (sqrt (- (/ (- 0.5 (/ 0.25 (* x_m x_m))) x_m) -0.5)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.05) {
		tmp = (0.375 * (x_m * x_m)) / fma(sqrt(2.0), sqrt(0.5), 2.0);
	} else {
		tmp = 1.0 - sqrt((((0.5 - (0.25 / (x_m * x_m))) / x_m) - -0.5));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.05)
		tmp = Float64(Float64(0.375 * Float64(x_m * x_m)) / fma(sqrt(2.0), sqrt(0.5), 2.0));
	else
		tmp = Float64(1.0 - sqrt(Float64(Float64(Float64(0.5 - Float64(0.25 / Float64(x_m * x_m))) / x_m) - -0.5)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.05], N[(N[(0.375 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[N[(N[(N[(0.5 - N[(0.25 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] - -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05000000000000004

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 34.8%

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

   7. Applied rewrites 34.9%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}^{1.5}}{\left(1 + \mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right) + \sqrt{\mathsf{fma}\left(x \cdot x, -0.25, 1\right)}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{3}{8} \cdot \frac{{x}^{2}}{2 + \sqrt{\frac{1}{2}} \cdot \sqrt{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}} \]

   if 1.05000000000000004 < 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

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

   5. Applied rewrites 97.3%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05:\\ \;\;\;\;\frac{0.375 \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(\sqrt{2}, \sqrt{0.5}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{0.5 - \frac{0.25}{x \cdot x}}{x} - -0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;1 - \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} - -1}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2e-77) (- 1.0 (sqrt 1.0)) (/ 0.5 (- (sqrt 0.5) -1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2e-77) {
		tmp = 1.0 - sqrt(1.0);
	} else {
		tmp = 0.5 / (sqrt(0.5) - -1.0);
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.2d-77) then
        tmp = 1.0d0 - sqrt(1.0d0)
    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 <= 2.2e-77) {
		tmp = 1.0 - Math.sqrt(1.0);
	} 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 <= 2.2e-77:
		tmp = 1.0 - math.sqrt(1.0)
	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 <= 2.2e-77)
		tmp = Float64(1.0 - sqrt(1.0));
	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 <= 2.2e-77)
		tmp = 1.0 - sqrt(1.0);
	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, 2.2e-77], N[(1.0 - N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[Sqrt[0.5], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 38.3%

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

   if 2.20000000000000007e-77 < x

   1. Initial program 82.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 79.2%

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.2%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;1 - \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{0.5} - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;1 - \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.2e-77) (- 1.0 (sqrt 1.0)) (- 1.0 (sqrt 0.5))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.2e-77) {
		tmp = 1.0 - sqrt(1.0);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.2d-77) then
        tmp = 1.0d0 - sqrt(1.0d0)
    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 <= 2.2e-77) {
		tmp = 1.0 - Math.sqrt(1.0);
	} 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 <= 2.2e-77:
		tmp = 1.0 - math.sqrt(1.0)
	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 <= 2.2e-77)
		tmp = Float64(1.0 - sqrt(1.0));
	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 <= 2.2e-77)
		tmp = 1.0 - sqrt(1.0);
	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, 2.2e-77], N[(1.0 - N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000007e-77

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 38.3%

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

   if 2.20000000000000007e-77 < x

   1. Initial program 82.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   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 78.1%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-77}:\\ \;\;\;\;1 - \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 - \sqrt{0.5} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (- 1.0 (sqrt 0.5)))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 - sqrt(0.5);
}

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = 1.0d0 - sqrt(0.5d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 - Math.sqrt(0.5);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 - math.sqrt(0.5)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 - sqrt(0.5))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 - sqrt(0.5);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
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 52.4%

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

6. Final simplification 52.4%

   \[\leadsto 1 - \sqrt{0.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))