Given's Rotation SVD example

Percentage Accurate: 79.2% → 99.8%

Time: 9.1s

Alternatives: 8

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: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.999999998:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{p\_m \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}, -1.5, p\_m\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(4, p\_m \cdot p\_m, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999999998)
   (/ (fma (/ (* p_m (* p_m p_m)) (* x x)) -1.5 p_m) (- x))
   (sqrt (fma (/ x (sqrt (fma 4.0 (* p_m p_m) (* x x)))) 0.5 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999999998) {
		tmp = fma(((p_m * (p_m * p_m)) / (x * x)), -1.5, p_m) / -x;
	} else {
		tmp = sqrt(fma((x / sqrt(fma(4.0, (p_m * p_m), (x * x)))), 0.5, 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.999999998)
		tmp = Float64(fma(Float64(Float64(p_m * Float64(p_m * p_m)) / Float64(x * x)), -1.5, p_m) / Float64(-x));
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(4.0, Float64(p_m * p_m), Float64(x * x)))), 0.5, 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.999999998], N[(N[(N[(N[(p$95$m * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -1.5 + p$95$m), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(4.0 * N[(p$95$m * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $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)))) < -0.999999997999999946

   1. Initial program 16.9%

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

      1. 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 16.9

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 16.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \sqrt{\color{blue}{\frac{\frac{1}{4} \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{{x}^{2}} + {p}^{2}}{{x}^{2}}}} \]
   6. 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. lower-/.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-sqr N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 43.3

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

   7. Simplified 43.3%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{p + \frac{-3}{2} \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. 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. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 54.4

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

   10. Simplified 54.4%

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

   if -0.999999997999999946 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 99.8

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999999998:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{p \cdot \left(p \cdot p\right)}{x \cdot x}, -1.5, p\right)}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(4, p \cdot p, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

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 x (/ 0.25 p_m) 0.5))
       (fma (/ (* p_m p_m) (* x x)) -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(x, (0.25 / p_m), 0.5));
	} else {
		tmp = fma(((p_m * p_m) / (x * x)), -0.5, 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(x, Float64(0.25 / p_m), 0.5));
	else
		tmp = fma(Float64(Float64(p_m * p_m) / Float64(x * x)), -0.5, 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[(x * N[(0.25 / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.5 + 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 17.6

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 17.6%

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

   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. Simplified 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. 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.2

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

   7. Simplified 98.2%

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

   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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 87.8%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

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 x (/ 0.25 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(x, (0.25 / 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(x, Float64(0.25 / 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[(x * N[(0.25 / 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 17.6

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 17.6%

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

   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. Simplified 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. 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 98.2

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

   7. Simplified 98.2%

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

   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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 99.3%

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

4. Final simplification 87.7%

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

Alternative 4: 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 17.6

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 17.6%

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

   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. Simplified 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

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

      1. lower-sqrt.f64 98.1

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

   5. Simplified 98.1%

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

   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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 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 5: 77.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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) 0.0 (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 = 0.0;
	} 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 = 0.0d0
    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 = 0.0;
	} 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 = 0.0
	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 = 0.0;
	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 = 0.0;
	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], 0.0, 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

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

   5. Simplified 16.4%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{-1}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{0}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\color{blue}{0}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

      4. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

      5. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(1 + -1\right)}^{\frac{1}{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(1 + -1\right)}}^{\frac{1}{2}} \]

      7. metadata-eval N/A

         \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

      8. metadata-eval 16.4

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{0} \]

   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

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

      1. lower-sqrt.f64 97.0

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

   5. Simplified 97.0%

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

   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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 100.0

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 97.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.6× speedup

Math

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

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)
   (/ (- 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 = -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(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / fma(Float64(p_m * 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[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(N[(p$95$m * p$95$m), $MachinePrecision] * N[(2.0 / x), $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. 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 17.6

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 17.6%

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

   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. Simplified 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))))

   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. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 87.5%

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

Alternative 7: 38.7% accurate, 1.3× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0) 0.0 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)))) <= -1.0) {
		tmp = 0.0;
	} 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)))) <= (-1.0d0)) then
        tmp = 0.0d0
    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)))) <= -1.0) {
		tmp = 0.0;
	} 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)))) <= -1.0:
		tmp = 0.0
	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)))) <= -1.0)
		tmp = 0.0;
	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)))) <= -1.0)
		tmp = 0.0;
	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], -1.0], 0.0, 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)))) < -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

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

   5. Simplified 16.4%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{-1}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{0}} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\color{blue}{0}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

      4. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

      5. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(1 + -1\right)}^{\frac{1}{2}}} \]

      6. lift-+.f64 N/A

         \[\leadsto {\color{blue}{\left(1 + -1\right)}}^{\frac{1}{2}} \]

      7. metadata-eval N/A

         \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

      8. metadata-eval 16.4

         \[\leadsto \color{blue}{0} \]

   7. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{0} \]

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

   1. Initial program 99.5%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sqrt.f64 99.5

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 37.7%

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

4. Final simplification 32.7%

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

Alternative 8: 6.3% accurate, 58.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 0 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 0.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 0.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 0.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 0.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 0.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 0.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 0.0

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. Taylor expanded in x around -inf

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

5. Simplified 6.2%

   \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{-1}\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{0}} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\color{blue}{0}} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

   4. lift-+.f64 N/A

      \[\leadsto \sqrt{\color{blue}{1 + -1}} \]

   5. pow1/2 N/A

      \[\leadsto \color{blue}{{\left(1 + -1\right)}^{\frac{1}{2}}} \]

   6. lift-+.f64 N/A

      \[\leadsto {\color{blue}{\left(1 + -1\right)}}^{\frac{1}{2}} \]

   7. metadata-eval N/A

      \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

   8. metadata-eval 6.2

      \[\leadsto \color{blue}{0} \]

7. Applied egg-rr 6.2%

   \[\leadsto \color{blue}{0} \]
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))))))))