Given's Rotation SVD example

Percentage Accurate: 78.7% → 99.6%

Time: 11.7s

Alternatives: 5

Speedup: 0.7×

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: 78.7% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{\left(p\_m \cdot \left(p\_m \cdot p\_m\right)\right) \cdot \frac{1.5}{x \cdot x} - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.5)
   (/ (- (* (* p_m (* p_m p_m)) (/ 1.5 (* x x))) p_m) x)
   (sqrt (+ 0.5 (/ (* x 0.5) (hypot x (* p_m 2.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5) {
		tmp = (((p_m * (p_m * p_m)) * (1.5 / (x * x))) - p_m) / x;
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5) {
		tmp = (((p_m * (p_m * p_m)) * (1.5 / (x * x))) - p_m) / x;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5:
		tmp = (((p_m * (p_m * p_m)) * (1.5 / (x * x))) - p_m) / x
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / math.hypot(x, (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.5)
		tmp = Float64(Float64(Float64(Float64(p_m * Float64(p_m * p_m)) * Float64(1.5 / Float64(x * x))) - p_m) / x);
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / hypot(x, Float64(p_m * 2.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.5)
		tmp = (((p_m * (p_m * p_m)) * (1.5 / (x * x))) - p_m) / x;
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(N[(N[(p$95$m * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - p$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 12.3%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 12.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 12.3%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{-1 \cdot \color{blue}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(p + \frac{1}{8} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right), \color{blue}{\left(-1 \cdot x\right)}\right) \]

   7. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{p + \frac{0.125 \cdot \left(\left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot -12\right)}{x \cdot \left(x \cdot p\right)}}{0 - x}} \]

   8. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot p + \frac{3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}\right), \color{blue}{x}\right) \]

   10. Simplified 42.1%

       \[\leadsto \color{blue}{\frac{\left(p \cdot \left(p \cdot p\right)\right) \cdot \frac{1.5}{x \cdot x} - p}{x}} \]

   if -0.5 < (/.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. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt{\frac{\frac{1}{2} - \frac{\frac{1}{2} \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}{\frac{1}{2} \cdot \frac{1}{2} - \frac{\frac{1}{2} \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}} \cdot \frac{\frac{1}{2} \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}}\right)}\right) \]

   6. Applied egg-rr 99.0%

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. remove-double-div N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{\frac{1}{2} \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{x \cdot x + 4 \cdot \left(p \cdot p\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{x \cdot x + \left(p \cdot p\right) \cdot 4}\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{x \cdot x + \left(p \cdot p\right) \cdot \left(2 \cdot 2\right)}\right)\right)\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{x \cdot x + \left(p \cdot 2\right) \cdot \left(p \cdot 2\right)}\right)\right)\right)\right) \]

      12. hypot-define N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\mathsf{hypot}\left(x, p \cdot 2\right)\right)\right)\right)\right) \]

      13. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{hypot.f64}\left(x, \left(p \cdot 2\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{hypot.f64}\left(x, \mathsf{*.f64}\left(p, 2\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 100.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{\left(p \cdot \left(p \cdot p\right)\right) \cdot \frac{1.5}{x \cdot x} - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.65 \cdot 10^{-263}:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{elif}\;p\_m \leq 2.05 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{p\_m \cdot 2 + x \cdot \left(x \cdot \frac{0.25}{p\_m}\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.65e-263)
   (- 0.0 (/ p_m x))
   (if (<= p_m 2.05e-13)
     1.0
     (sqrt (+ 0.5 (/ (* x 0.5) (+ (* p_m 2.0) (* x (* x (/ 0.25 p_m))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.65e-263) {
		tmp = 0.0 - (p_m / x);
	} else if (p_m <= 2.05e-13) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / ((p_m * 2.0) + (x * (x * (0.25 / p_m)))))));
	}
	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 (p_m <= 1.65d-263) then
        tmp = 0.0d0 - (p_m / x)
    else if (p_m <= 2.05d-13) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + ((x * 0.5d0) / ((p_m * 2.0d0) + (x * (x * (0.25d0 / p_m)))))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.65e-263) {
		tmp = 0.0 - (p_m / x);
	} else if (p_m <= 2.05e-13) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / ((p_m * 2.0) + (x * (x * (0.25 / p_m)))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.65e-263:
		tmp = 0.0 - (p_m / x)
	elif p_m <= 2.05e-13:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / ((p_m * 2.0) + (x * (x * (0.25 / p_m)))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.65e-263)
		tmp = Float64(0.0 - Float64(p_m / x));
	elseif (p_m <= 2.05e-13)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / Float64(Float64(p_m * 2.0) + Float64(x * Float64(x * Float64(0.25 / p_m)))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.65e-263)
		tmp = 0.0 - (p_m / x);
	elseif (p_m <= 2.05e-13)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + ((x * 0.5) / ((p_m * 2.0) + (x * (x * (0.25 / p_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.65e-263], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 2.05e-13], 1.0, N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[(N[(p$95$m * 2.0), $MachinePrecision] + N[(x * N[(x * N[(0.25 / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.6499999999999998e-263

   1. Initial program 73.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 73.1%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p}{x}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{p}{x}\right)}\right) \]

      4. /-lowering-/.f64 8.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(p, \color{blue}{x}\right)\right) \]

   7. Simplified 8.2%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p}{x}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{p}{x}\right)\right) \]

      3. /-lowering-/.f64 8.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(p, x\right)\right) \]

   9. Applied egg-rr 8.2%

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

   if 1.6499999999999998e-263 < p < 2.0500000000000001e-13

   1. Initial program 66.9%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 66.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 54.0%

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

   if 2.0500000000000001e-13 < p

   1. Initial program 98.7%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 98.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 98.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \color{blue}{\left(\frac{1}{4} \cdot \frac{{x}^{2}}{p} + 2 \cdot p\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(2 \cdot p + \frac{1}{4} \cdot \frac{{x}^{2}}{p}\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(2 \cdot p + \frac{\frac{1}{4} \cdot {x}^{2}}{p}\right)\right)\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(2 \cdot p + \frac{\frac{1}{4}}{p} \cdot {x}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(2 \cdot p + \frac{\frac{1}{4} \cdot 1}{p} \cdot {x}^{2}\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(2 \cdot p + \left(\frac{1}{4} \cdot \frac{1}{p}\right) \cdot {x}^{2}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\left(2 \cdot p\right), \left(\left(\frac{1}{4} \cdot \frac{1}{p}\right) \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\left(p \cdot 2\right), \left(\left(\frac{1}{4} \cdot \frac{1}{p}\right) \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(\left(\frac{1}{4} \cdot \frac{1}{p}\right) \cdot {x}^{2}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left({x}^{2} \cdot \left(\frac{1}{4} \cdot \frac{1}{p}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(\left(x \cdot x\right) \cdot \left(\frac{1}{4} \cdot \frac{1}{p}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(x \cdot \left(x \cdot \left(\frac{1}{4} \cdot \frac{1}{p}\right)\right)\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(x \cdot \left(x \cdot \frac{\frac{1}{4} \cdot 1}{p}\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(x \cdot \left(x \cdot \frac{\frac{1}{4}}{p}\right)\right)\right)\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(x \cdot \frac{x \cdot \frac{1}{4}}{p}\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(x \cdot \frac{\frac{1}{4} \cdot x}{p}\right)\right)\right)\right)\right) \]

      16. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \left(x \cdot \left(\frac{1}{4} \cdot \frac{x}{p}\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \frac{x}{p}\right)\right)\right)\right)\right)\right) \]

      18. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{4} \cdot x}{p}\right)\right)\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \left(\frac{x \cdot \frac{1}{4}}{p}\right)\right)\right)\right)\right)\right) \]

      20. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \left(x \cdot \frac{\frac{1}{4}}{p}\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \left(x \cdot \frac{\frac{1}{4} \cdot 1}{p}\right)\right)\right)\right)\right)\right) \]

      22. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{4} \cdot \frac{1}{p}\right)\right)\right)\right)\right)\right)\right) \]

      23. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \frac{1}{p}\right)\right)\right)\right)\right)\right)\right) \]

      24. associate-*r/ N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{4} \cdot 1}{p}\right)\right)\right)\right)\right)\right)\right) \]

      25. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{4}}{p}\right)\right)\right)\right)\right)\right)\right) \]

      26. /-lowering-/.f64 97.6%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(p, 2\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{4}, p\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 97.6%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.65 \cdot 10^{-263}:\\ \;\;\;\;0 - \frac{p}{x}\\ \mathbf{elif}\;p \leq 2.05 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{p \cdot 2 + x \cdot \left(x \cdot \frac{0.25}{p}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.58 \cdot 10^{-263}:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{elif}\;p\_m \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.58e-263)
   (- 0.0 (/ p_m x))
   (if (<= p_m 2.9e-13) 1.0 (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.58e-263) {
		tmp = 0.0 - (p_m / x);
	} else if (p_m <= 2.9e-13) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	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 (p_m <= 1.58d-263) then
        tmp = 0.0d0 - (p_m / x)
    else if (p_m <= 2.9d-13) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.58e-263) {
		tmp = 0.0 - (p_m / x);
	} else if (p_m <= 2.9e-13) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.58e-263:
		tmp = 0.0 - (p_m / x)
	elif p_m <= 2.9e-13:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.58e-263)
		tmp = Float64(0.0 - Float64(p_m / x));
	elseif (p_m <= 2.9e-13)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.58e-263)
		tmp = 0.0 - (p_m / x);
	elseif (p_m <= 2.9e-13)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.58e-263], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 2.9e-13], 1.0, N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.57999999999999998e-263

   1. Initial program 73.1%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 73.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 73.1%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p}{x}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{p}{x}\right)}\right) \]

      4. /-lowering-/.f64 8.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(p, \color{blue}{x}\right)\right) \]

   7. Simplified 8.2%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p}{x}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{p}{x}\right)\right) \]

      3. /-lowering-/.f64 8.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(p, x\right)\right) \]

   9. Applied egg-rr 8.2%

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

   if 1.57999999999999998e-263 < p < 2.8999999999999998e-13

   1. Initial program 66.9%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 66.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 66.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 54.0%

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

   if 2.8999999999999998e-13 < p

   1. Initial program 98.7%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 98.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 98.7%

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

   5. Taylor expanded in x around 0

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

      1. sqrt-lowering-sqrt.f64 97.6%

         \[\leadsto \mathsf{sqrt.f64}\left(\frac{1}{2}\right) \]

   7. Simplified 97.6%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.58 \cdot 10^{-263}:\\ \;\;\;\;0 - \frac{p}{x}\\ \mathbf{elif}\;p \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.4% accurate, 21.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{-150}:\\ \;\;\;\;0 - \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -3.75e-150) (- 0.0 (/ p_m x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.75e-150) {
		tmp = 0.0 - (p_m / x);
	} 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 (x <= (-3.75d-150)) then
        tmp = 0.0d0 - (p_m / x)
    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 (x <= -3.75e-150) {
		tmp = 0.0 - (p_m / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.75e-150:
		tmp = 0.0 - (p_m / x)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.75e-150)
		tmp = Float64(0.0 - Float64(p_m / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -3.75e-150)
		tmp = 0.0 - (p_m / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.75e-150], N[(0.0 - N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.7500000000000002e-150

   1. Initial program 56.5%

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

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 56.5%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 56.5%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p}{x}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{p}{x}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{p}{x}\right)}\right) \]

      4. /-lowering-/.f64 22.7%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(p, \color{blue}{x}\right)\right) \]

   7. Simplified 22.7%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{p}{x}\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{p}{x}\right)\right) \]

      3. /-lowering-/.f64 22.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(p, x\right)\right) \]

   9. Applied egg-rr 22.7%

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

   if -3.7500000000000002e-150 < 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. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      2. distribute-rgt-in N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 57.2%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{-150}:\\ \;\;\;\;0 - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.9% accurate, 215.0× 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 79.4%

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

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(1 \cdot \frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \left(\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}\right)\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\left(\left(4 \cdot p\right) \cdot p + x \cdot x\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(\left(4 \cdot p\right) \cdot p\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(4 \cdot \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \left(p \cdot p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 79.4%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(p, p\right)\right), \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

3. Simplified 79.4%

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

5. Taylor expanded in x around inf

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

7. Simplified 36.1%

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

Developer Target 1: 78.7% accurate, 0.7× 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 2024160 
(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))))))))