Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.8%

Time: 11.5s

Alternatives: 11

Speedup: 0.6×

• could not determine a ground truth

  1. p = -9.721987841690694e-156
  2. x = -5.8468203097034966e+113

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% 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, 0.2× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := p\_m \cdot \left(4 \cdot p\_m\right)\\ t_1 := \frac{x \cdot 0.5}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}\\ \mathbf{if}\;\frac{x}{\sqrt{t\_0 + x \cdot x}} \leq -0.9999:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_1}^{3} + 0.125}{\mathsf{fma}\left(t\_1, t\_1, 0.25 - 0.5 \cdot t\_1\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* p_m (* 4.0 p_m))) (t_1 (/ (* x 0.5) (sqrt (fma x x t_0)))))
   (if (<= (/ x (sqrt (+ t_0 (* x x)))) -0.9999)
     (- (/ p_m x))
     (sqrt (/ (+ (pow t_1 3.0) 0.125) (fma t_1 t_1 (- 0.25 (* 0.5 t_1))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m * (4.0 * p_m);
	double t_1 = (x * 0.5) / sqrt(fma(x, x, t_0));
	double tmp;
	if ((x / sqrt((t_0 + (x * x)))) <= -0.9999) {
		tmp = -(p_m / x);
	} else {
		tmp = sqrt(((pow(t_1, 3.0) + 0.125) / fma(t_1, t_1, (0.25 - (0.5 * t_1)))));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m * Float64(4.0 * p_m))
	t_1 = Float64(Float64(x * 0.5) / sqrt(fma(x, x, t_0)))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(t_0 + Float64(x * x)))) <= -0.9999)
		tmp = Float64(-Float64(p_m / x));
	else
		tmp = sqrt(Float64(Float64((t_1 ^ 3.0) + 0.125) / fma(t_1, t_1, Float64(0.25 - Float64(0.5 * t_1)))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999], (-N[(p$95$m / x), $MachinePrecision]), N[Sqrt[N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + 0.125), $MachinePrecision] / N[(t$95$1 * t$95$1 + N[(0.25 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99990000000000001

   1. Initial program 12.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.9

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

   5. Simplified 57.9%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.7

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

   7. Applied egg-rr 58.7%

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

   if -0.99990000000000001 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

   4. Applied egg-rr 99.9%

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

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. flip3-+ N/A

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

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

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

   6. Applied egg-rr 99.9%

      \[\leadsto \sqrt{\color{blue}{\frac{{\left(\frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}\right)}^{3} + 0.125}{\mathsf{fma}\left(\frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}, \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}}, 0.25 - \frac{0.5 \cdot x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(p \cdot 4\right)\right)}} \cdot 0.5\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\frac{x \cdot 0.5}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}\right)}^{3} + 0.125}{\mathsf{fma}\left(\frac{x \cdot 0.5}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}, \frac{x \cdot 0.5}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}, 0.25 - 0.5 \cdot \frac{x \cdot 0.5}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.6:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + \frac{p\_m \cdot p\_m}{x \cdot x}}}\\ \end{array} \end{array} \]

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 -0.6)
     (- (/ p_m x))
     (if (<= t_0 5e-6)
       (sqrt (fma 0.25 (/ x p_m) 0.5))
       (/ 1.0 (sqrt (+ 1.0 (/ (* p_m p_m) (* x x)))))))))

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 <= -0.6) {
		tmp = -(p_m / x);
	} else if (t_0 <= 5e-6) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = 1.0 / sqrt((1.0 + ((p_m * p_m) / (x * x))));
	}
	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 <= -0.6)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 5e-6)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(p_m * p_m) / Float64(x * x)))));
	end
	return 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, -0.6], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 5e-6], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.599999999999999978

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.5

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

   5. Simplified 57.5%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   if -0.599999999999999978 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 5.00000000000000041e-6

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if 5.00000000000000041e-6 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

   4. Applied egg-rr 100.0%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

      6. associate-/l/ N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      13. accelerator-lowering-fma.f64 N/A

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

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

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.7

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

   9. Simplified 99.7%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{1 + \frac{p \cdot p}{x \cdot x}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{1 + \frac{p \cdot p}{x \cdot x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.6:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -0.5 \cdot \left(p\_m \cdot p\_m\right)\right)}{x \cdot x}\\ \end{array} \end{array} \]

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 -0.6)
     (- (/ p_m x))
     (if (<= t_0 5e-6)
       (sqrt (fma 0.25 (/ x p_m) 0.5))
       (/ (fma x x (* -0.5 (* p_m p_m))) (* x x))))))

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 <= -0.6) {
		tmp = -(p_m / x);
	} else if (t_0 <= 5e-6) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = fma(x, x, (-0.5 * (p_m * p_m))) / (x * x);
	}
	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 <= -0.6)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 5e-6)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = Float64(fma(x, x, Float64(-0.5 * Float64(p_m * p_m))) / Float64(x * x));
	end
	return 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, -0.6], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 5e-6], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(x * x + N[(-0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.599999999999999978

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.5

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

   5. Simplified 57.5%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   if -0.599999999999999978 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 5.00000000000000041e-6

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if 5.00000000000000041e-6 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 99.7

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

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

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

       5. *-commutative N/A

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

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

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

       7. unpow2 N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 99.7

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

   12. Simplified 99.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -0.5 \cdot \left(p \cdot p\right)\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.6:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p\_m \cdot p\_m}{x \cdot x}, 1\right)\\ \end{array} \end{array} \]

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 -0.6)
     (- (/ p_m x))
     (if (<= t_0 5e-6)
       (sqrt (fma 0.25 (/ x p_m) 0.5))
       (fma -0.5 (/ (* p_m p_m) (* x x)) 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 <= -0.6) {
		tmp = -(p_m / x);
	} else if (t_0 <= 5e-6) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = fma(-0.5, ((p_m * p_m) / (x * x)), 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 <= -0.6)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 5e-6)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = fma(-0.5, Float64(Float64(p_m * p_m) / Float64(x * x)), 1.0);
	end
	return 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, -0.6], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 5e-6], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.599999999999999978

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.5

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

   5. Simplified 57.5%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   if -0.599999999999999978 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 5.00000000000000041e-6

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if 5.00000000000000041e-6 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 99.7

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

   9. Simplified 99.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p \cdot p}{x \cdot x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.6:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.6)
     (- (/ p_m x))
     (if (<= t_0 5e-6) (sqrt (fma 0.25 (/ x p_m) 0.5)) 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 <= -0.6) {
		tmp = -(p_m / x);
	} else if (t_0 <= 5e-6) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = 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 <= -0.6)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 5e-6)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(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, -0.6], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 5e-6], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.599999999999999978

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.5

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

   5. Simplified 57.5%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   if -0.599999999999999978 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 5.00000000000000041e-6

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. /-lowering-/.f64 98.1

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

   5. Simplified 98.1%

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

   if 5.00000000000000041e-6 < (/.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. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

      1. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := p\_m \cdot \left(4 \cdot p\_m\right)\\ \mathbf{if}\;\frac{x}{\sqrt{t\_0 + x \cdot x}} \leq -0.9999:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\frac{1}{1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}}}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* p_m (* 4.0 p_m))))
   (if (<= (/ x (sqrt (+ t_0 (* x x)))) -0.9999)
     (- (/ p_m x))
     (sqrt (/ 0.5 (/ 1.0 (+ 1.0 (/ x (sqrt (fma x x t_0))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m * (4.0 * p_m);
	double tmp;
	if ((x / sqrt((t_0 + (x * x)))) <= -0.9999) {
		tmp = -(p_m / x);
	} else {
		tmp = sqrt((0.5 / (1.0 / (1.0 + (x / sqrt(fma(x, x, t_0)))))));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m * Float64(4.0 * p_m))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(t_0 + Float64(x * x)))) <= -0.9999)
		tmp = Float64(-Float64(p_m / x));
	else
		tmp = sqrt(Float64(0.5 / Float64(1.0 / Float64(1.0 + Float64(x / sqrt(fma(x, x, t_0)))))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999], (-N[(p$95$m / x), $MachinePrecision]), N[Sqrt[N[(0.5 / N[(1.0 / N[(1.0 + N[(x / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99990000000000001

   1. Initial program 12.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.9

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

   5. Simplified 57.9%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.7

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

   7. Applied egg-rr 58.7%

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

   if -0.99990000000000001 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\frac{1}{1 + \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.6:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.6) (- (/ p_m x)) (if (<= t_0 5e-6) (sqrt 0.5) 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 <= -0.6) {
		tmp = -(p_m / x);
	} else if (t_0 <= 5e-6) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / sqrt(((p_m * (4.0d0 * p_m)) + (x * x)))
    if (t_0 <= (-0.6d0)) then
        tmp = -(p_m / x)
    else if (t_0 <= 5d-6) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.6) {
		tmp = -(p_m / x);
	} else if (t_0 <= 5e-6) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 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 <= -0.6:
		tmp = -(p_m / x)
	elif t_0 <= 5e-6:
		tmp = math.sqrt(0.5)
	else:
		tmp = 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 <= -0.6)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 5e-6)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -0.6)
		tmp = -(p_m / x);
	elseif (t_0 <= 5e-6)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(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, -0.6], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 5e-6], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.599999999999999978

   1. Initial program 14.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.5

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

   5. Simplified 57.5%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.3

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

   7. Applied egg-rr 58.3%

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

   if -0.599999999999999978 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 5.00000000000000041e-6

   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. Taylor expanded in x around 0

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

   5. Simplified 98.0%

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

   if 5.00000000000000041e-6 < (/.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. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

      1. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.6:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := p\_m \cdot \left(4 \cdot p\_m\right)\\ \mathbf{if}\;\frac{x}{\sqrt{t\_0 + x \cdot x}} \leq -0.9999:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (* p_m (* 4.0 p_m))))
   (if (<= (/ x (sqrt (+ t_0 (* x x)))) -0.9999)
     (- (/ p_m x))
     (sqrt (fma (/ x (sqrt (fma x x t_0))) 0.5 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m * (4.0 * p_m);
	double tmp;
	if ((x / sqrt((t_0 + (x * x)))) <= -0.9999) {
		tmp = -(p_m / x);
	} else {
		tmp = sqrt(fma((x / sqrt(fma(x, x, t_0))), 0.5, 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m * Float64(4.0 * p_m))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(t_0 + Float64(x * x)))) <= -0.9999)
		tmp = Float64(-Float64(p_m / x));
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(x, x, t_0))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999], (-N[(p$95$m / x), $MachinePrecision]), N[Sqrt[N[(N[(x / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99990000000000001

   1. Initial program 12.8%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 57.9

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

   5. Simplified 57.9%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 58.7

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

   7. Applied egg-rr 58.7%

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

   if -0.99990000000000001 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. neg-lowering-neg.f64 56.8

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

   5. Simplified 56.8%

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

      1. associate-/l* N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. sqrt-unprod N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. frac-2neg N/A

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

      9. remove-double-neg N/A

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

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

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

      11. neg-lowering-neg.f64 57.6

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

   7. Applied egg-rr 57.6%

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

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

   1. Initial program 100.0%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

   4. Applied egg-rr 100.0%

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

      1. associate-/r/ N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 99.0

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

   9. Simplified 99.0%

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

4. Final simplification 90.2%

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

Alternative 10: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) 5e-6) (sqrt 0.5) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= 5e-6) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / sqrt(((p_m * (4.0d0 * p_m)) + (x * x)))) <= 5d-6) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= 5e-6) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= 5e-6:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= 5e-6)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= 5e-6)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-6], N[Sqrt[0.5], $MachinePrecision], 1.0]

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)))) < 5.00000000000000041e-6

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 71.9%

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

   if 5.00000000000000041e-6 < (/.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. Taylor expanded in x around inf

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

   5. Simplified 99.3%

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

      1. metadata-eval 99.3

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.2% accurate, 58.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in x around inf

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

5. Simplified 38.8%

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

   1. metadata-eval 38.8

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

7. Applied egg-rr 38.8%

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

Developer Target 1: 79.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024201 
(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))))))))