Given's Rotation SVD example

Percentage Accurate: 80.6% → 99.8%

Time: 11.2s

Alternatives: 6

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: 80.6% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{0 - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(t\_0 + 1\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -1.0) (/ (- 0.0 p_m) x) (sqrt (* 0.5 (+ t_0 1.0))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (0.0 - p_m) / x;
	} else {
		tmp = sqrt((0.5 * (t_0 + 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 = x / sqrt(((p_m * (4.0d0 * p_m)) + (x * x)))
    if (t_0 <= (-1.0d0)) then
        tmp = (0.0d0 - p_m) / x
    else
        tmp = sqrt((0.5d0 * (t_0 + 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 = x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (0.0 - p_m) / x;
	} else {
		tmp = Math.sqrt((0.5 * (t_0 + 1.0)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))
	tmp = 0
	if t_0 <= -1.0:
		tmp = (0.0 - p_m) / x
	else:
		tmp = math.sqrt((0.5 * (t_0 + 1.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(N[(0.0 - p$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(t$95$0 + 1.0), $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)))) < -1

   1. Initial program 14.7%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 14.7%

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

   3. Simplified 14.7%

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

   5. Taylor expanded in x around -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 57.7%

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

   7. Simplified 57.7%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   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 57.7%

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

   9. Applied egg-rr 57.7%

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

   if -1 < (/.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. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{0 - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -155000000:\\ \;\;\;\;\frac{0 - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{x + p\_m \cdot \left(p\_m \cdot \left(\frac{\left(p\_m \cdot p\_m\right) \cdot -2}{x \cdot \left(x \cdot x\right)} + \frac{2}{x}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -155000000.0)
   (/ (- 0.0 p_m) x)
   (sqrt
    (+
     0.5
     (*
      x
      (/
       0.5
       (+
        x
        (*
         p_m
         (* p_m (+ (/ (* (* p_m p_m) -2.0) (* x (* x x))) (/ 2.0 x)))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -155000000.0) {
		tmp = (0.0 - p_m) / x;
	} else {
		tmp = sqrt((0.5 + (x * (0.5 / (x + (p_m * (p_m * ((((p_m * p_m) * -2.0) / (x * (x * x))) + (2.0 / x)))))))));
	}
	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 <= (-155000000.0d0)) then
        tmp = (0.0d0 - p_m) / x
    else
        tmp = sqrt((0.5d0 + (x * (0.5d0 / (x + (p_m * (p_m * ((((p_m * p_m) * (-2.0d0)) / (x * (x * x))) + (2.0d0 / x)))))))))
    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 <= -155000000.0) {
		tmp = (0.0 - p_m) / x;
	} else {
		tmp = Math.sqrt((0.5 + (x * (0.5 / (x + (p_m * (p_m * ((((p_m * p_m) * -2.0) / (x * (x * x))) + (2.0 / x)))))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -155000000.0:
		tmp = (0.0 - p_m) / x
	else:
		tmp = math.sqrt((0.5 + (x * (0.5 / (x + (p_m * (p_m * ((((p_m * p_m) * -2.0) / (x * (x * x))) + (2.0 / x)))))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -155000000.0)
		tmp = Float64(Float64(0.0 - p_m) / x);
	else
		tmp = sqrt(Float64(0.5 + Float64(x * Float64(0.5 / Float64(x + Float64(p_m * Float64(p_m * Float64(Float64(Float64(Float64(p_m * p_m) * -2.0) / Float64(x * Float64(x * x))) + Float64(2.0 / x)))))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -155000000.0)
		tmp = (0.0 - p_m) / x;
	else
		tmp = sqrt((0.5 + (x * (0.5 / (x + (p_m * (p_m * ((((p_m * p_m) * -2.0) / (x * (x * x))) + (2.0 / x)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -155000000.0], N[(N[(0.0 - p$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(x * N[(0.5 / N[(x + N[(p$95$m * N[(p$95$m * N[(N[(N[(N[(p$95$m * p$95$m), $MachinePrecision] * -2.0), $MachinePrecision] / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55e8

   1. Initial program 45.2%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 45.2%

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

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}} \]
   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 39.6%

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

   7. Simplified 39.6%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   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 39.6%

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

   9. Applied egg-rr 39.6%

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

   if -1.55e8 < x

   1. Initial program 90.4%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in p around 0

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), \color{blue}{\left(x + {p}^{2} \cdot \left(-2 \cdot \frac{{p}^{2}}{{x}^{3}} + 2 \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

      3. unpow2 N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. /-lowering-/.f64 83.4%

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

   7. Simplified 83.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -155000000:\\ \;\;\;\;\frac{0 - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + x \cdot \frac{0.5}{x + p \cdot \left(p \cdot \left(\frac{\left(p \cdot p\right) \cdot -2}{x \cdot \left(x \cdot x\right)} + \frac{2}{x}\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 9.5 \cdot 10^{-262}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{0 - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{p\_m \cdot 2 + x \cdot \left(x \cdot \frac{0.25}{p\_m}\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 9.5e-262)
   1.0
   (if (<= p_m 4.8e-27)
     (/ (- 0.0 p_m) x)
     (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 <= 9.5e-262) {
		tmp = 1.0;
	} else if (p_m <= 4.8e-27) {
		tmp = (0.0 - p_m) / x;
	} 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 <= 9.5d-262) then
        tmp = 1.0d0
    else if (p_m <= 4.8d-27) then
        tmp = (0.0d0 - p_m) / x
    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 <= 9.5e-262) {
		tmp = 1.0;
	} else if (p_m <= 4.8e-27) {
		tmp = (0.0 - p_m) / x;
	} 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 <= 9.5e-262:
		tmp = 1.0
	elif p_m <= 4.8e-27:
		tmp = (0.0 - p_m) / x
	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 <= 9.5e-262)
		tmp = 1.0;
	elseif (p_m <= 4.8e-27)
		tmp = Float64(Float64(0.0 - p_m) / x);
	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 <= 9.5e-262)
		tmp = 1.0;
	elseif (p_m <= 4.8e-27)
		tmp = (0.0 - p_m) / x;
	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, 9.5e-262], 1.0, If[LessEqual[p$95$m, 4.8e-27], N[(N[(0.0 - p$95$m), $MachinePrecision] / x), $MachinePrecision], 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 < 9.4999999999999999e-262

   1. Initial program 78.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 34.3%

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

   if 9.4999999999999999e-262 < p < 4.80000000000000004e-27

   1. Initial program 60.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   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 50.2%

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

   7. Simplified 50.2%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   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 50.2%

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

   9. Applied egg-rr 50.2%

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

   if 4.80000000000000004e-27 < p

   1. Initial program 92.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around 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 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 9.5 \cdot 10^{-262}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{0 - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{p \cdot 2 + x \cdot \left(x \cdot \frac{0.25}{p}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.3 \cdot 10^{-259}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 5.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{0 - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.3e-259) 1.0 (if (<= p_m 5.5e-28) (/ (- 0.0 p_m) x) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.3e-259) {
		tmp = 1.0;
	} else if (p_m <= 5.5e-28) {
		tmp = (0.0 - p_m) / x;
	} 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 <= 1.3d-259) then
        tmp = 1.0d0
    else if (p_m <= 5.5d-28) then
        tmp = (0.0d0 - p_m) / x
    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 <= 1.3e-259) {
		tmp = 1.0;
	} else if (p_m <= 5.5e-28) {
		tmp = (0.0 - p_m) / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.3e-259:
		tmp = 1.0
	elif p_m <= 5.5e-28:
		tmp = (0.0 - p_m) / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.3e-259)
		tmp = 1.0;
	elseif (p_m <= 5.5e-28)
		tmp = Float64(Float64(0.0 - p_m) / x);
	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 <= 1.3e-259)
		tmp = 1.0;
	elseif (p_m <= 5.5e-28)
		tmp = (0.0 - p_m) / x;
	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, 1.3e-259], 1.0, If[LessEqual[p$95$m, 5.5e-28], N[(N[(0.0 - p$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.3e-259

   1. Initial program 78.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 34.3%

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

   if 1.3e-259 < p < 5.49999999999999967e-28

   1. Initial program 60.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x}} \]
   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 50.2%

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

   7. Simplified 50.2%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   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 50.2%

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

   9. Applied egg-rr 50.2%

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

   if 5.49999999999999967e-28 < p

   1. Initial program 92.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around 0

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

      1. sqrt-lowering-sqrt.f64 89.7%

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

   7. Simplified 89.7%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.3 \cdot 10^{-259}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{0 - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.9% accurate, 21.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-146}:\\ \;\;\;\;\frac{0 - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2e-146) (/ (- 0.0 p_m) x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2e-146) {
		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 <= (-2d-146)) 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 <= -2e-146) {
		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 <= -2e-146:
		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 <= -2e-146)
		tmp = Float64(Float64(0.0 - 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 <= -2e-146)
		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, -2e-146], N[(N[(0.0 - p$95$m), $MachinePrecision] / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000005e-146

   1. Initial program 58.6%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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 58.6%

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

   3. Simplified 58.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{0.5 \cdot x}{\sqrt{4 \cdot \left(p \cdot p\right) + x \cdot x}}}} \]
   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 29.4%

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

   7. Simplified 29.4%

      \[\leadsto \color{blue}{0 - \frac{p}{x}} \]
   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 29.4%

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

   9. Applied egg-rr 29.4%

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

   if -2.00000000000000005e-146 < 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 58.1%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-146}:\\ \;\;\;\;\frac{0 - p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.3% 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 78.4%

   \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
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 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in x around inf

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

7. Simplified 34.4%

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

Developer Target 1: 80.6% 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 2024161 
(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))))))))