Given's Rotation SVD example

Percentage Accurate: 79.2% → 99.9%

Time: 10.7s

Alternatives: 9

Speedup: 0.6×

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: 79.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := p\_m \cdot \left(4 \cdot p\_m\right)\\ \mathbf{if}\;\frac{x}{\sqrt{t\_0 + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(p\_m \cdot -1.5, \frac{p\_m \cdot p\_m}{x \cdot x}, p\_m\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(x, x, t\_0\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

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)))) -0.5)
     (/ (fma (* p_m -1.5) (/ (* p_m p_m) (* x x)) 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)))) <= -0.5) {
		tmp = fma((p_m * -1.5), ((p_m * p_m) / (x * x)), 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)))) <= -0.5)
		tmp = Float64(fma(Float64(p_m * -1.5), Float64(Float64(p_m * p_m) / Float64(x * x)), p_m) / Float64(-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], -0.5], N[(N[(N[(p$95$m * -1.5), $MachinePrecision] * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + p$95$m), $MachinePrecision] / (-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)))) < -0.5

   1. Initial program 17.6%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{\color{blue}{-12}}{{x}^{2}}, {p}^{2}\right)}{{x}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{\color{blue}{x \cdot x}}, {p}^{2}\right)}{{x}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{\color{blue}{x \cdot x}}, {p}^{2}\right)}{{x}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, \color{blue}{p \cdot p}\right)}{{x}^{2}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, \color{blue}{p \cdot p}\right)}{{x}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

      20. lower-*.f64 42.6

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(0.25, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

   5. Applied rewrites 42.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{fma}\left(0.25, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{x \cdot x}}} \]

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{\color{blue}{-1 \cdot x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{-1 \cdot x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}} + p}}{-1 \cdot x} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}, p\right)}{-1 \cdot x} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}, p\right)}{-1 \cdot x} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, p\right)}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      16. lower-neg.f64 53.5

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

   8. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, p\right)}{-x}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{p \cdot \color{blue}{\left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{\color{blue}{p \cdot \left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{\color{blue}{p \cdot \left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\left(p \cdot \frac{p \cdot p}{x \cdot x}\right)} + p}{\mathsf{neg}\left(x\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(p \cdot \color{blue}{\frac{p \cdot p}{x \cdot x}}\right) + p}{\mathsf{neg}\left(x\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot p\right) \cdot \frac{p \cdot p}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{2} \cdot p, \frac{p \cdot p}{x \cdot x}, p\right)}}{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

   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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 89.1%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m \cdot p\_m}{x \cdot x}\\ t_1 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(p\_m \cdot -1.5, t\_0, p\_m\right)}{-x}\\ \mathbf{elif}\;t\_1 \leq 0.6:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (* p_m p_m) (* x x)))
        (t_1 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_1 -0.5)
     (/ (fma (* p_m -1.5) t_0 p_m) (- x))
     (if (<= t_1 0.6) (sqrt (fma 0.25 (/ x p_m) 0.5)) (fma -0.5 t_0 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = (p_m * p_m) / (x * x);
	double t_1 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma((p_m * -1.5), t_0, p_m) / -x;
	} else if (t_1 <= 0.6) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = fma(-0.5, t_0, 1.0);
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(p_m * p_m) / Float64(x * x))
	t_1 = Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = Float64(fma(Float64(p_m * -1.5), t_0, p_m) / Float64(-x));
	elseif (t_1 <= 0.6)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = fma(-0.5, t_0, 1.0);
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$1, -0.5], N[(N[(N[(p$95$m * -1.5), $MachinePrecision] * t$95$0 + p$95$m), $MachinePrecision] / (-x)), $MachinePrecision], If[LessEqual[t$95$1, 0.6], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(-0.5 * t$95$0 + 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)))) < -0.5

   1. Initial program 17.6%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{\color{blue}{-12}}{{x}^{2}}, {p}^{2}\right)}{{x}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{\color{blue}{x \cdot x}}, {p}^{2}\right)}{{x}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{\color{blue}{x \cdot x}}, {p}^{2}\right)}{{x}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, \color{blue}{p \cdot p}\right)}{{x}^{2}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, \color{blue}{p \cdot p}\right)}{{x}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

      20. lower-*.f64 42.6

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(0.25, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

   5. Applied rewrites 42.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{fma}\left(0.25, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{x \cdot x}}} \]

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{\color{blue}{-1 \cdot x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{-1 \cdot x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}} + p}}{-1 \cdot x} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}, p\right)}{-1 \cdot x} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}, p\right)}{-1 \cdot x} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, p\right)}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      16. lower-neg.f64 53.5

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

   8. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, p\right)}{-x}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{p \cdot \color{blue}{\left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{\color{blue}{p \cdot \left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{\color{blue}{p \cdot \left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\left(p \cdot \frac{p \cdot p}{x \cdot x}\right)} + p}{\mathsf{neg}\left(x\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(p \cdot \color{blue}{\frac{p \cdot p}{x \cdot x}}\right) + p}{\mathsf{neg}\left(x\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot p\right) \cdot \frac{p \cdot p}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{2} \cdot p, \frac{p \cdot p}{x \cdot x}, p\right)}}{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 0.599999999999999978

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.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. lower-*.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. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(p \cdot -1.5, \frac{p \cdot p}{x \cdot x}, p\right)}{-x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.6:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p \cdot p}{x \cdot x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

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 -0.5)
     (- (/ p_m x))
     (if (<= t_0 0.6)
       (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 <= -0.5) {
		tmp = -(p_m / x);
	} else if (t_0 <= 0.6) {
		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 <= -0.5)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 0.6)
		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, -0.5], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 0.6], 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)))) < -0.5

   1. Initial program 17.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 17.6

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

   4. Applied rewrites 17.6%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 54.2

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

   7. Applied rewrites 54.2%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 0.599999999999999978

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.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. lower-*.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. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 87.8%

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

Alternative 4: 99.2% accurate, 0.5× speedup

Math

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

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 -0.5)
     (- (/ p_m x))
     (if (<= t_0 0.6) (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 <= -0.5) {
		tmp = -(p_m / x);
	} else if (t_0 <= 0.6) {
		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 <= -0.5)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 0.6)
		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, -0.5], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 0.6], 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)))) < -0.5

   1. Initial program 17.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 17.6

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

   4. Applied rewrites 17.6%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 54.2

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

   7. Applied rewrites 54.2%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 0.599999999999999978

   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. lower-fma.f64 N/A

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

      3. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

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

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.3%

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

4. Final simplification 87.7%

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

Alternative 5: 98.4% accurate, 0.6× speedup

Math

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

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 -0.5) (- (/ p_m x)) (if (<= t_0 4e-5) (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 <= -0.5) {
		tmp = -(p_m / x);
	} else if (t_0 <= 4e-5) {
		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 <= (-0.5d0)) then
        tmp = -(p_m / x)
    else if (t_0 <= 4d-5) 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 <= -0.5) {
		tmp = -(p_m / x);
	} else if (t_0 <= 4e-5) {
		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 <= -0.5:
		tmp = -(p_m / x)
	elif t_0 <= 4e-5:
		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 <= -0.5)
		tmp = Float64(-Float64(p_m / x));
	elseif (t_0 <= 4e-5)
		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 <= -0.5)
		tmp = -(p_m / x);
	elseif (t_0 <= 4e-5)
		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, -0.5], (-N[(p$95$m / x), $MachinePrecision]), If[LessEqual[t$95$0, 4e-5], 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)))) < -0.5

   1. Initial program 17.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 17.6

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

   4. Applied rewrites 17.6%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 54.2

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

   7. Applied rewrites 54.2%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000033e-5

   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. Applied rewrites 98.1%

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

   if 4.00000000000000033e-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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 97.6%

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

4. Final simplification 87.4%

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

Alternative 6: 77.1% accurate, 0.6× speedup

Math

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

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 -1.0) (/ p_m x) (if (<= t_0 4e-5) (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 <= -1.0) {
		tmp = p_m / x;
	} else if (t_0 <= 4e-5) {
		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 <= (-1.0d0)) then
        tmp = p_m / x
    else if (t_0 <= 4d-5) 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 <= -1.0) {
		tmp = p_m / x;
	} else if (t_0 <= 4e-5) {
		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 <= -1.0:
		tmp = p_m / x
	elif t_0 <= 4e-5:
		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 <= -1.0)
		tmp = Float64(p_m / x);
	elseif (t_0 <= 4e-5)
		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 <= -1.0)
		tmp = p_m / x;
	elseif (t_0 <= 4e-5)
		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, -1.0], N[(p$95$m / x), $MachinePrecision], If[LessEqual[t$95$0, 4e-5], 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)))) < -1

   1. Initial program 16.4%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in p around 0

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

      1. lower-/.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000033e-5

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.0%

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

   if 4.00000000000000033e-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. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 97.6%

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

4. Final simplification 88.7%

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

Alternative 7: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(p\_m \cdot -1.5, \frac{p\_m \cdot p\_m}{x \cdot x}, p\_m\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{fma}\left(p\_m, p\_m \cdot \frac{2}{x}, x\right)}\right)}\\ \end{array} \end{array} \]

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)
   (/ (fma (* p_m -1.5) (/ (* p_m p_m) (* x x)) p_m) (- x))
   (sqrt (* 0.5 (+ 1.0 (/ x (fma p_m (* p_m (/ 2.0 x)) x)))))))

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 = fma((p_m * -1.5), ((p_m * p_m) / (x * x)), p_m) / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / fma(p_m, (p_m * (2.0 / x)), x)))));
	}
	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(fma(Float64(p_m * -1.5), Float64(Float64(p_m * p_m) / Float64(x * x)), p_m) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / fma(p_m, Float64(p_m * Float64(2.0 / x)), x)))));
	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[(N[(N[(p$95$m * -1.5), $MachinePrecision] * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + p$95$m), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(p$95$m * N[(p$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + x), $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)))) < -0.5

   1. Initial program 17.6%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-rgt-out N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{\color{blue}{-12}}{{x}^{2}}, {p}^{2}\right)}{{x}^{2}}} \]

      15. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{\color{blue}{x \cdot x}}, {p}^{2}\right)}{{x}^{2}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{\color{blue}{x \cdot x}}, {p}^{2}\right)}{{x}^{2}}} \]

      17. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, \color{blue}{p \cdot p}\right)}{{x}^{2}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, \color{blue}{p \cdot p}\right)}{{x}^{2}}} \]

      19. unpow2 N/A

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\frac{1}{4}, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

      20. lower-*.f64 42.6

          \[\leadsto \sqrt{\frac{\mathsf{fma}\left(0.25, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{\color{blue}{x \cdot x}}} \]

   5. Applied rewrites 42.6%

      \[\leadsto \sqrt{\color{blue}{\frac{\mathsf{fma}\left(0.25, \left(\left(p \cdot p\right) \cdot \left(p \cdot p\right)\right) \cdot \frac{-12}{x \cdot x}, p \cdot p\right)}{x \cdot x}}} \]

   6. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{\color{blue}{-1 \cdot x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{-1 \cdot x}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}} + p}}{-1 \cdot x} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}, p\right)}{-1 \cdot x} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}}, p\right)}{-1 \cdot x} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-3}{2}, \frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, p\right)}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      16. lower-neg.f64 53.5

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

   8. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.5, \frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, p\right)}{-x}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{p \cdot \color{blue}{\left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{\color{blue}{p \cdot \left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{p \cdot \left(p \cdot p\right)}{\color{blue}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \frac{\color{blue}{p \cdot \left(p \cdot p\right)}}{x \cdot x} + p}{\mathsf{neg}\left(x\right)} \]

      7. associate-/l* N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \color{blue}{\left(p \cdot \frac{p \cdot p}{x \cdot x}\right)} + p}{\mathsf{neg}\left(x\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{-3}{2} \cdot \left(p \cdot \color{blue}{\frac{p \cdot p}{x \cdot x}}\right) + p}{\mathsf{neg}\left(x\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-3}{2} \cdot p\right) \cdot \frac{p \cdot p}{x \cdot x}} + p}{\mathsf{neg}\left(x\right)} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{2} \cdot p, \frac{p \cdot p}{x \cdot x}, p\right)}}{\mathsf{neg}\left(x\right)} \]

      11. lower-*.f64 55.1

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

   10. Applied rewrites 55.1%

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

   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. Taylor expanded in p around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 98.1

          \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{fma}\left(p, p \cdot \color{blue}{\frac{2}{x}}, x\right)}\right)} \]

   5. Applied rewrites 98.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(p \cdot -1.5, \frac{p \cdot p}{x \cdot x}, p\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{fma}\left(p, p \cdot \frac{2}{x}, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq 0.45:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) 0.45) (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)))) <= 0.45) {
		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)))) <= 0.45d0) 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)))) <= 0.45) {
		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)))) <= 0.45:
		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)))) <= 0.45)
		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)))) <= 0.45)
		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], 0.45], 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)))) < 0.450000000000000011

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.8%

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

   if 0.450000000000000011 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 97.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.45:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.1% accurate, 58.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 80.0

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

4. Applied rewrites 80.0%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 30.0%

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

Developer Target 1: 79.2% 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 2024219 
(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))))))))