Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.9%

Time: 8.5s

Alternatives: 6

Speedup: 0.5×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))

C

double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p) * p) + (x * x)))))))
end function

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p) * p) + (x * x)))))));
}

Python

def code(p, x):
	return math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p) * p) + (x * x)))))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p) * p) + Float64(x * x)))))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p), $MachinePrecision] * p), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \left(4 \cdot p\_m\right) \cdot p\_m\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{t\_0 + x \cdot x}}\right)} \leq 10^{-5}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* (* 4.0 p_m) p_m)))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ t_0 (* x x))))))) 1e-5)
     (/ (- p_m) x)
     (exp (* (log (fma (/ x (sqrt (fma x x t_0))) 0.5 0.5)) 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = (4.0 * p_m) * p_m;
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((t_0 + (x * x))))))) <= 1e-5) {
		tmp = -p_m / x;
	} else {
		tmp = exp((log(fma((x / sqrt(fma(x, x, t_0))), 0.5, 0.5)) * 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(4.0 * p_m) * p_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(t_0 + Float64(x * x))))))) <= 1e-5)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = exp(Float64(log(fma(Float64(x / sqrt(fma(x, x, t_0))), 0.5, 0.5)) * 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-5], N[((-p$95$m) / x), $MachinePrecision], N[Exp[N[(N[Log[N[(N[(x / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 1.00000000000000008e-5

   1. Initial program 13.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 50.2

          \[\leadsto \left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{\sqrt{0.5}}}{x}\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\sqrt{0.5}}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if 1.00000000000000008e-5 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.8%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}\right)} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 10^{-5}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.7071067811865476:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.2)
     (/ (- p_m) x)
     (if (<= t_0 0.7071067811865476) (sqrt (fma (/ x p_m) 0.25 0.5)) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.7071067811865476) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.2)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.7071067811865476)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.2], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.7071067811865476], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.20000000000000001

   1. Initial program 15.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 49.3

          \[\leadsto \left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{\sqrt{0.5}}}{x}\right) \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\sqrt{0.5}}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if 0.20000000000000001 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.707106781186547573

   1. Initial program 99.8%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 0.707106781186547573 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}\right)} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in p around 0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.2:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.7071067811865476:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.7071067811865476:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0
         (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))))
   (if (<= t_0 0.2)
     (/ (- p_m) x)
     (if (<= t_0 0.7071067811865476) (sqrt 0.5) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.7071067811865476) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p_m) * p_m) + (x * x)))))))
    if (t_0 <= 0.2d0) then
        tmp = -p_m / x
    else if (t_0 <= 0.7071067811865476d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	double tmp;
	if (t_0 <= 0.2) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.7071067811865476) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p_m) * p_m) + (x * x)))))))
	tmp = 0
	if t_0 <= 0.2:
		tmp = -p_m / x
	elif t_0 <= 0.7071067811865476:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x)))))))
	tmp = 0.0
	if (t_0 <= 0.2)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.7071067811865476)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x)))))));
	tmp = 0.0;
	if (t_0 <= 0.2)
		tmp = -p_m / x;
	elseif (t_0 <= 0.7071067811865476)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.2], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.7071067811865476], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.20000000000000001

   1. Initial program 15.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 49.3

          \[\leadsto \left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{\sqrt{0.5}}}{x}\right) \]

   5. Applied rewrites 49.3%

      \[\leadsto \color{blue}{\left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\sqrt{0.5}}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.7%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if 0.20000000000000001 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.707106781186547573

   1. Initial program 99.8%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 97.8%

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

   if 0.707106781186547573 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}\right)} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in p around 0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.2:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 0.7071067811865476:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \left(4 \cdot p\_m\right) \cdot p\_m\\ \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{t\_0 + x \cdot x}}\right)} \leq 10^{-5}:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* (* 4.0 p_m) p_m)))
   (if (<= (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ t_0 (* x x))))))) 1e-5)
     (/ (- p_m) x)
     (sqrt (fma (/ x (sqrt (fma x x t_0))) 0.5 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = (4.0 * p_m) * p_m;
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((t_0 + (x * x))))))) <= 1e-5) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / sqrt(fma(x, x, t_0))), 0.5, 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(4.0 * p_m) * p_m)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(t_0 + Float64(x * x))))))) <= 1e-5)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(x, x, t_0))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-5], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 1.00000000000000008e-5

   1. Initial program 13.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 50.2

          \[\leadsto \left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\color{blue}{\sqrt{0.5}}}{x}\right) \]

   5. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\left(-p\right) \cdot \left(\sqrt{2} \cdot \frac{\sqrt{0.5}}{x}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{-p}{x}} \]

   if 1.00000000000000008e-5 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 99.8%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}\right)} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 99.9

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}, 0.5, 0.5\right)}} \]

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.9

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(4 \cdot p\right)} \cdot p\right)}}, 0.5, 0.5\right)} \]

   6. Applied rewrites 99.9%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \leq 10^{-5}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\_m\right) \cdot p\_m + x \cdot x}}\right)} \leq 0.7071067811865476:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<=
      (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))))))
      0.7071067811865476)
   (sqrt 0.5)
   1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.7071067811865476) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (sqrt((0.5d0 * (1.0d0 + (x / sqrt((((4.0d0 * p_m) * p_m) + (x * x))))))) <= 0.7071067811865476d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (Math.sqrt((0.5 * (1.0 + (x / Math.sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.7071067811865476) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if math.sqrt((0.5 * (1.0 + (x / math.sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.7071067811865476:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / sqrt(Float64(Float64(Float64(4.0 * p_m) * p_m) + Float64(x * x))))))) <= 0.7071067811865476)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p_m) * p_m) + (x * x))))))) <= 0.7071067811865476)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.7071067811865476], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))))) < 0.707106781186547573

   1. Initial program 66.2%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in p around inf

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

   5. Applied rewrites 60.8%

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

   if 0.707106781186547573 < (sqrt.f64 (*.f64 #s(literal 1/2 binary64) (+.f64 #s(literal 1 binary64) (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))))))

   1. Initial program 100.0%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 100.0

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}\right)} \]

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}} \]

   7. Taylor expanded in p around 0

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

   9. Applied rewrites 100.0%

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

Alternative 6: 36.1% accurate, 58.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 1 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 1.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 1.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 1.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 1.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 76.3%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 76.3

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}}\right)} \]

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 76.3

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, \color{blue}{\left(p \cdot 4\right)} \cdot p\right)}}\right)} \]

4. Applied rewrites 76.3%

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\mathsf{fma}\left(x, x, \left(p \cdot 4\right) \cdot p\right)}}}\right)} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. pow-to-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-*.f64 N/A

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

6. Applied rewrites 76.3%

   \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, \left(4 \cdot p\right) \cdot p\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}} \]

7. Taylor expanded in p around 0

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

9. Applied rewrites 39.6%

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

Developer Target 1: 79.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024337 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))