Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.6%

Time: 13.0s

Alternatives: 9

Speedup: 1.0×

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: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p\_m + -1.5 \cdot \frac{{p\_m}^{3}}{{x}^{2}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.98)
   (/ (+ p_m (* -1.5 (/ (pow p_m 3.0) (pow x 2.0)))) (- x))
   (pow
    (pow (fma x (/ 0.5 (hypot x (* p_m 2.0))) 0.5) 1.5)
    0.3333333333333333)))

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)))) <= -0.98) {
		tmp = (p_m + (-1.5 * (pow(p_m, 3.0) / pow(x, 2.0)))) / -x;
	} else {
		tmp = pow(pow(fma(x, (0.5 / hypot(x, (p_m * 2.0))), 0.5), 1.5), 0.3333333333333333);
	}
	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)))) <= -0.98)
		tmp = Float64(Float64(p_m + Float64(-1.5 * Float64((p_m ^ 3.0) / (x ^ 2.0)))) / Float64(-x));
	else
		tmp = (fma(x, Float64(0.5 / hypot(x, Float64(p_m * 2.0))), 0.5) ^ 1.5) ^ 0.3333333333333333;
	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], -0.98], N[(N[(p$95$m + N[(-1.5 * N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Power[N[Power[N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], 1.5], $MachinePrecision], 0.3333333333333333], $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)))) < -0.97999999999999998

   1. Initial program 12.8%

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

      1. +-commutative 12.8%

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

      2. associate-*l* 12.8%

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

      3. fma-define 12.8%

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

      4. sqr-neg 12.8%

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

      5. fma-define 12.8%

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

      6. associate-*l* 12.8%

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

      7. +-commutative 12.8%

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

      8. distribute-lft-in 12.8%

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

      9. metadata-eval 12.8%

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

   3. Simplified 12.8%

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

      1. associate-*r/ 12.8%

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

      2. clear-num 12.8%

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

      3. fma-undefine 12.8%

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

      4. associate-*r* 12.8%

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

      5. add-sqr-sqrt 12.8%

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

      6. hypot-define 12.8%

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

      7. associate-*r* 12.8%

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

      8. *-commutative 12.8%

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

      9. sqrt-prod 12.8%

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

      10. sqrt-prod 5.6%

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

      11. add-sqr-sqrt 12.8%

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

      12. metadata-eval 12.8%

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

      13. *-commutative 12.8%

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

   6. Applied egg-rr 12.8%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x \cdot 0.5}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 12.9%

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

      2. associate-*r* 12.9%

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

      3. associate-*l/ 12.8%

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

      4. *-lft-identity 12.8%

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

      5. metadata-eval 12.8%

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

      6. times-frac 12.8%

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

      7. *-rgt-identity 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in x around -inf 39.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 39.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right)}{x}} \]

       2. mul-1-neg 39.8%

          \[\leadsto \frac{\color{blue}{-\left(p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right)}}{x} \]

       3. distribute-rgt-out 39.8%

          \[\leadsto \frac{-\left(p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}\right)}{x} \]

       4. metadata-eval 39.8%

          \[\leadsto \frac{-\left(p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}\right)}{x} \]

   11. Simplified 39.8%

       \[\leadsto \color{blue}{\frac{-\left(p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}\right)}{x}} \]

   12. Taylor expanded in p around 0 47.5%

       \[\leadsto \frac{-\left(p + \color{blue}{-1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}\right)}{x} \]

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-define 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. fma-undefine 99.9%

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

      4. associate-*r* 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. hypot-define 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. sqrt-prod 100.0%

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

      10. sqrt-prod 52.1%

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

      11. add-sqr-sqrt 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x \cdot 0.5}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. metadata-eval 100.0%

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

      6. times-frac 100.0%

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

      7. *-rgt-identity 100.0%

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

   8. Simplified 100.0%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}} \cdot \sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right) \cdot \sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 100.0%

         \[\leadsto {\left(\color{blue}{\left(0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)} \cdot \sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}^{0.3333333333333333} \]

      4. pow1 100.0%

         \[\leadsto {\left(\color{blue}{{\left(0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{1}} \cdot \sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}^{0.3333333333333333} \]

      5. pow1/2 100.0%

         \[\leadsto {\left({\left(0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{1} \cdot \color{blue}{{\left(0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 100.0%

         \[\leadsto {\color{blue}{\left({\left(0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. +-commutative 100.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 0.5\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. associate-/l* 100.0%

         \[\leadsto {\left({\left(\color{blue}{x \cdot \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}} + 0.5\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. fma-define 100.0%

         \[\leadsto {\left({\color{blue}{\left(\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      10. metadata-eval 100.0%

          \[\leadsto {\left({\left(\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p\_m + -1.5 \cdot \frac{{p\_m}^{3}}{{x}^{2}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.98)
   (/ (+ p_m (* -1.5 (/ (pow p_m 3.0) (pow x 2.0)))) (- x))
   (sqrt (fma x (/ 0.5 (hypot x (* p_m 2.0))) 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)))) <= -0.98) {
		tmp = (p_m + (-1.5 * (pow(p_m, 3.0) / pow(x, 2.0)))) / -x;
	} else {
		tmp = sqrt(fma(x, (0.5 / hypot(x, (p_m * 2.0))), 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)))) <= -0.98)
		tmp = Float64(Float64(p_m + Float64(-1.5 * Float64((p_m ^ 3.0) / (x ^ 2.0)))) / Float64(-x));
	else
		tmp = sqrt(fma(x, Float64(0.5 / hypot(x, Float64(p_m * 2.0))), 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], -0.98], N[(N[(p$95$m + N[(-1.5 * N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 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)))) < -0.97999999999999998

   1. Initial program 12.8%

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

      1. +-commutative 12.8%

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

      2. associate-*l* 12.8%

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

      3. fma-define 12.8%

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

      4. sqr-neg 12.8%

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

      5. fma-define 12.8%

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

      6. associate-*l* 12.8%

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

      7. +-commutative 12.8%

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

      8. distribute-lft-in 12.8%

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

      9. metadata-eval 12.8%

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

   3. Simplified 12.8%

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

      1. associate-*r/ 12.8%

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

      2. clear-num 12.8%

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

      3. fma-undefine 12.8%

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

      4. associate-*r* 12.8%

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

      5. add-sqr-sqrt 12.8%

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

      6. hypot-define 12.8%

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

      7. associate-*r* 12.8%

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

      8. *-commutative 12.8%

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

      9. sqrt-prod 12.8%

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

      10. sqrt-prod 5.6%

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

      11. add-sqr-sqrt 12.8%

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

      12. metadata-eval 12.8%

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

      13. *-commutative 12.8%

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

   6. Applied egg-rr 12.8%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x \cdot 0.5}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 12.9%

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

      2. associate-*r* 12.9%

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

      3. associate-*l/ 12.8%

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

      4. *-lft-identity 12.8%

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

      5. metadata-eval 12.8%

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

      6. times-frac 12.8%

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

      7. *-rgt-identity 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in x around -inf 39.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 39.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right)}{x}} \]

       2. mul-1-neg 39.8%

          \[\leadsto \frac{\color{blue}{-\left(p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right)}}{x} \]

       3. distribute-rgt-out 39.8%

          \[\leadsto \frac{-\left(p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}\right)}{x} \]

       4. metadata-eval 39.8%

          \[\leadsto \frac{-\left(p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}\right)}{x} \]

   11. Simplified 39.8%

       \[\leadsto \color{blue}{\frac{-\left(p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}\right)}{x}} \]

   12. Taylor expanded in p around 0 47.5%

       \[\leadsto \frac{-\left(p + \color{blue}{-1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}\right)}{x} \]

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-define 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 66.4%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 66.4%

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

      2. *-commutative 66.4%

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

      3. *-un-lft-identity 66.4%

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

      4. times-frac 66.4%

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

      5. metadata-eval 66.4%

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

      6. metadata-eval 66.4%

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

      7. unpow2 66.4%

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

      8. flip-+ 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-+r+ 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. associate-+l+ 100.0%

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

      14. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{1 \cdot \sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}} \]
   9. Step-by-step derivation

      1. *-lft-identity 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r/ 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r/ 100.0%

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

      6. metadata-eval 100.0%

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

      7. fma-undefine 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p\_m + -1.5 \cdot \frac{{p\_m}^{3}}{{x}^{2}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.98)
   (/ (+ p_m (* -1.5 (/ (pow p_m 3.0) (pow x 2.0)))) (- x))
   (sqrt (+ 0.5 (/ (* x 0.5) (hypot x (* p_m 2.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98) {
		tmp = (p_m + (-1.5 * (pow(p_m, 3.0) / pow(x, 2.0)))) / -x;
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98) {
		tmp = (p_m + (-1.5 * (Math.pow(p_m, 3.0) / Math.pow(x, 2.0)))) / -x;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98:
		tmp = (p_m + (-1.5 * (math.pow(p_m, 3.0) / math.pow(x, 2.0)))) / -x
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / math.hypot(x, (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.98)
		tmp = Float64(Float64(p_m + Float64(-1.5 * Float64((p_m ^ 3.0) / (x ^ 2.0)))) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / hypot(x, Float64(p_m * 2.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98)
		tmp = (p_m + (-1.5 * ((p_m ^ 3.0) / (x ^ 2.0)))) / -x;
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(p$95$m + N[(-1.5 * N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.97999999999999998

   1. Initial program 12.8%

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

      1. +-commutative 12.8%

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

      2. associate-*l* 12.8%

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

      3. fma-define 12.8%

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

      4. sqr-neg 12.8%

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

      5. fma-define 12.8%

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

      6. associate-*l* 12.8%

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

      7. +-commutative 12.8%

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

      8. distribute-lft-in 12.8%

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

      9. metadata-eval 12.8%

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

   3. Simplified 12.8%

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

      1. associate-*r/ 12.8%

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

      2. clear-num 12.8%

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

      3. fma-undefine 12.8%

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

      4. associate-*r* 12.8%

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

      5. add-sqr-sqrt 12.8%

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

      6. hypot-define 12.8%

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

      7. associate-*r* 12.8%

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

      8. *-commutative 12.8%

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

      9. sqrt-prod 12.8%

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

      10. sqrt-prod 5.6%

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

      11. add-sqr-sqrt 12.8%

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

      12. metadata-eval 12.8%

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

      13. *-commutative 12.8%

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

   6. Applied egg-rr 12.8%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x \cdot 0.5}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 12.9%

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

      2. associate-*r* 12.9%

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

      3. associate-*l/ 12.8%

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

      4. *-lft-identity 12.8%

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

      5. metadata-eval 12.8%

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

      6. times-frac 12.8%

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

      7. *-rgt-identity 12.8%

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

   8. Simplified 12.8%

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

   9. Taylor expanded in x around -inf 39.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 39.8%

          \[\leadsto \color{blue}{\frac{-1 \cdot \left(p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right)}{x}} \]

       2. mul-1-neg 39.8%

          \[\leadsto \frac{\color{blue}{-\left(p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}\right)}}{x} \]

       3. distribute-rgt-out 39.8%

          \[\leadsto \frac{-\left(p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}\right)}{x} \]

       4. metadata-eval 39.8%

          \[\leadsto \frac{-\left(p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}\right)}{x} \]

   11. Simplified 39.8%

       \[\leadsto \color{blue}{\frac{-\left(p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}\right)}{x}} \]

   12. Taylor expanded in p around 0 47.5%

       \[\leadsto \frac{-\left(p + \color{blue}{-1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}\right)}{x} \]

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-define 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. fma-undefine 99.9%

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

      4. associate-*r* 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. hypot-define 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. sqrt-prod 100.0%

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

      10. sqrt-prod 52.1%

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

      11. add-sqr-sqrt 100.0%

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

      12. metadata-eval 100.0%

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

      13. *-commutative 100.0%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x \cdot 0.5}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. metadata-eval 100.0%

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

      6. times-frac 100.0%

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

      7. *-rgt-identity 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -740:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -740.0)
   (/ p_m (- x))
   (sqrt (+ 0.5 (/ (* x 0.5) (hypot x (* p_m 2.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -740.0) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -740.0) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -740.0:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / math.hypot(x, (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -740.0)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / hypot(x, Float64(p_m * 2.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -740.0)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -740.0], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -740

   1. Initial program 48.6%

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

      1. +-commutative 48.6%

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

      2. associate-*l* 48.6%

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

      3. fma-define 48.6%

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

      4. sqr-neg 48.6%

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

      5. fma-define 48.6%

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

      6. associate-*l* 48.6%

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

      7. +-commutative 48.6%

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

      8. distribute-lft-in 48.6%

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

      9. metadata-eval 48.6%

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

   3. Simplified 48.6%

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

   5. Taylor expanded in x around -inf 32.3%

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

      1. mul-1-neg 32.3%

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

      2. distribute-neg-frac2 32.3%

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

   7. Simplified 32.3%

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

   if -740 < x

   1. Initial program 87.0%

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

      1. +-commutative 87.0%

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

      2. associate-*l* 87.0%

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

      3. fma-define 87.0%

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

      4. sqr-neg 87.0%

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

      5. fma-define 87.0%

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

      6. associate-*l* 87.0%

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

      7. +-commutative 87.0%

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

      8. distribute-lft-in 87.0%

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

      9. metadata-eval 87.0%

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

   3. Simplified 87.0%

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

      1. associate-*r/ 87.0%

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

      2. clear-num 87.0%

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

      3. fma-undefine 87.0%

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

      4. associate-*r* 87.0%

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

      5. add-sqr-sqrt 87.0%

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

      6. hypot-define 87.0%

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

      7. associate-*r* 87.0%

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

      8. *-commutative 87.0%

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

      9. sqrt-prod 87.0%

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

      10. sqrt-prod 45.7%

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

      11. add-sqr-sqrt 87.0%

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

      12. metadata-eval 87.0%

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

      13. *-commutative 87.0%

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

   6. Applied egg-rr 87.0%

      \[\leadsto \sqrt{0.5 + \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x \cdot 0.5}}}} \]
   7. Step-by-step derivation

      1. associate-/r/ 87.0%

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

      2. associate-*r* 87.0%

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

      3. associate-*l/ 87.0%

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

      4. *-lft-identity 87.0%

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

      5. metadata-eval 87.0%

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

      6. times-frac 87.0%

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

      7. *-rgt-identity 87.0%

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

   8. Simplified 87.0%

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

Alternative 5: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 3.4 \cdot 10^{-183}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 4.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\left|\frac{x}{p\_m}\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 3.4e-183)
   1.0
   (if (<= p_m 4.8e-68) (/ 1.0 (fabs (/ x p_m))) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.4e-183) {
		tmp = 1.0;
	} else if (p_m <= 4.8e-68) {
		tmp = 1.0 / fabs((x / p_m));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 3.4d-183) then
        tmp = 1.0d0
    else if (p_m <= 4.8d-68) then
        tmp = 1.0d0 / abs((x / p_m))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 3.4e-183) {
		tmp = 1.0;
	} else if (p_m <= 4.8e-68) {
		tmp = 1.0 / Math.abs((x / p_m));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 3.4e-183:
		tmp = 1.0
	elif p_m <= 4.8e-68:
		tmp = 1.0 / math.fabs((x / p_m))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 3.4e-183)
		tmp = 1.0;
	elseif (p_m <= 4.8e-68)
		tmp = Float64(1.0 / abs(Float64(x / p_m)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 3.4e-183)
		tmp = 1.0;
	elseif (p_m <= 4.8e-68)
		tmp = 1.0 / abs((x / p_m));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 3.4e-183], 1.0, If[LessEqual[p$95$m, 4.8e-68], N[(1.0 / N[Abs[N[(x / p$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.40000000000000014e-183

   1. Initial program 74.3%

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

      1. +-commutative 74.3%

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

      2. associate-*l* 74.3%

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

      3. fma-define 74.3%

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

      4. sqr-neg 74.3%

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

      5. fma-define 74.3%

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

      6. associate-*l* 74.3%

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

      7. +-commutative 74.3%

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

      8. distribute-lft-in 74.3%

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

      9. metadata-eval 74.3%

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

   3. Simplified 74.3%

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

   6. Applied egg-rr 44.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 43.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)\right)} \]

      2. expm1-undefine 43.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)} - 1} \]

   8. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. log1p-undefine 73.7%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)}} - 1 \]

      2. rem-exp-log 73.7%

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

      3. distribute-lft-in 73.7%

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

      4. metadata-eval 73.7%

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

      5. fma-define 73.7%

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

   10. Applied egg-rr 73.7%

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

   11. Taylor expanded in x around inf 39.7%

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

   if 3.40000000000000014e-183 < p < 4.79999999999999982e-68

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

      2. associate-*l* 52.4%

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

      3. fma-define 52.4%

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

      4. sqr-neg 52.4%

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

      5. fma-define 52.4%

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

      6. associate-*l* 52.4%

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

      7. +-commutative 52.4%

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

      8. distribute-lft-in 52.4%

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

      9. metadata-eval 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in x around -inf 52.5%

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

      1. mul-1-neg 52.5%

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

      2. distribute-neg-frac2 52.5%

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

   7. Simplified 52.5%

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

      1. clear-num 52.2%

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

      2. inv-pow 52.2%

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

      3. add-sqr-sqrt 51.2%

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

      4. sqrt-unprod 53.6%

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

      5. sqr-neg 53.6%

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

      6. sqrt-unprod 2.2%

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

      7. add-sqr-sqrt 4.2%

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

   9. Applied egg-rr 4.2%

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

       1. unpow-1 4.2%

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

   11. Simplified 4.2%

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

       1. add-sqr-sqrt 2.2%

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

       2. sqrt-unprod 37.1%

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

       3. pow2 37.1%

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

   13. Applied egg-rr 37.1%

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

       1. unpow2 37.1%

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

       2. rem-sqrt-square 53.6%

          \[\leadsto \frac{1}{\color{blue}{\left|\frac{x}{p}\right|}} \]

   15. Simplified 53.6%

       \[\leadsto \frac{1}{\color{blue}{\left|\frac{x}{p}\right|}} \]

   if 4.79999999999999982e-68 < p

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 81.1%

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

Alternative 6: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.8 \cdot 10^{-183}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.8e-183) 1.0 (if (<= p_m 1.6e-66) (/ p_m (- x)) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.8e-183) {
		tmp = 1.0;
	} else if (p_m <= 1.6e-66) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 2.8d-183) then
        tmp = 1.0d0
    else if (p_m <= 1.6d-66) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.8e-183) {
		tmp = 1.0;
	} else if (p_m <= 1.6e-66) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.8e-183:
		tmp = 1.0
	elif p_m <= 1.6e-66:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.8e-183)
		tmp = 1.0;
	elseif (p_m <= 1.6e-66)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 2.8e-183)
		tmp = 1.0;
	elseif (p_m <= 1.6e-66)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 2.8e-183], 1.0, If[LessEqual[p$95$m, 1.6e-66], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.79999999999999985e-183

   1. Initial program 74.3%

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

      1. +-commutative 74.3%

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

      2. associate-*l* 74.3%

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

      3. fma-define 74.3%

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

      4. sqr-neg 74.3%

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

      5. fma-define 74.3%

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

      6. associate-*l* 74.3%

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

      7. +-commutative 74.3%

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

      8. distribute-lft-in 74.3%

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

      9. metadata-eval 74.3%

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

   3. Simplified 74.3%

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

   6. Applied egg-rr 44.0%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 43.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)\right)} \]

      2. expm1-undefine 43.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)} - 1} \]

   8. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. log1p-undefine 73.7%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)}} - 1 \]

      2. rem-exp-log 73.7%

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

      3. distribute-lft-in 73.7%

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

      4. metadata-eval 73.7%

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

      5. fma-define 73.7%

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

   10. Applied egg-rr 73.7%

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

   11. Taylor expanded in x around inf 39.7%

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

   if 2.79999999999999985e-183 < p < 1.59999999999999991e-66

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

      2. associate-*l* 52.4%

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

      3. fma-define 52.4%

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

      4. sqr-neg 52.4%

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

      5. fma-define 52.4%

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

      6. associate-*l* 52.4%

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

      7. +-commutative 52.4%

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

      8. distribute-lft-in 52.4%

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

      9. metadata-eval 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in x around -inf 52.5%

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

      1. mul-1-neg 52.5%

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

      2. distribute-neg-frac2 52.5%

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

   7. Simplified 52.5%

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

   if 1.59999999999999991e-66 < p

   1. Initial program 91.5%

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

   3. Taylor expanded in x around 0 81.1%

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

Alternative 7: 54.6% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -9.5e-211) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -9.5e-211) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-9.5d-211)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -9.5e-211) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -9.5e-211:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -9.5e-211)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -9.5e-211)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -9.5e-211], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -9.50000000000000008e-211

   1. Initial program 55.4%

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

      1. +-commutative 55.4%

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

      2. associate-*l* 55.4%

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

      3. fma-define 55.4%

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

      4. sqr-neg 55.4%

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

      5. fma-define 55.4%

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

      6. associate-*l* 55.4%

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

      7. +-commutative 55.4%

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

      8. distribute-lft-in 55.4%

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

      9. metadata-eval 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in x around -inf 25.9%

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

      1. mul-1-neg 25.9%

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

      2. distribute-neg-frac2 25.9%

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

   7. Simplified 25.9%

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

   if -9.50000000000000008e-211 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-define 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. associate-*l* 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 48.7%

      \[\leadsto \sqrt{\color{blue}{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 48.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)\right)} \]

      2. expm1-undefine 48.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)} - 1} \]

   8. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. log1p-undefine 99.3%

         \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)}} - 1 \]

      2. rem-exp-log 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. fma-define 99.3%

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

   10. Applied egg-rr 99.3%

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

   11. Taylor expanded in x around inf 61.1%

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

Alternative 8: 35.1% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.8%

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

   1. +-commutative 76.8%

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

   2. associate-*l* 76.8%

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

   3. fma-define 76.8%

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

   4. sqr-neg 76.8%

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

   5. fma-define 76.8%

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

   6. associate-*l* 76.8%

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

   7. +-commutative 76.8%

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

   8. distribute-lft-in 76.8%

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

   9. metadata-eval 76.8%

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

3. Simplified 76.8%

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

6. Applied egg-rr 52.2%

   \[\leadsto \sqrt{\color{blue}{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 51.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)\right)} \]

   2. expm1-undefine 51.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)} - 1} \]

8. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. log1p-undefine 76.1%

      \[\leadsto e^{\color{blue}{\log \left(1 + \sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)}} - 1 \]

   2. rem-exp-log 76.1%

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

   3. distribute-lft-in 76.1%

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

   4. metadata-eval 76.1%

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

   5. fma-define 76.1%

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

10. Applied egg-rr 76.1%

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

11. Taylor expanded in x around inf 35.9%

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

Alternative 9: 6.2% accurate, 215.0× 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 76.8%

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

   1. +-commutative 76.8%

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

   2. associate-*l* 76.8%

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

   3. fma-define 76.8%

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

   4. sqr-neg 76.8%

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

   5. fma-define 76.8%

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

   6. associate-*l* 76.8%

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

   7. +-commutative 76.8%

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

   8. distribute-lft-in 76.8%

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

   9. metadata-eval 76.8%

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

3. Simplified 76.8%

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

6. Applied egg-rr 52.2%

   \[\leadsto \sqrt{\color{blue}{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 51.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)\right)} \]

   2. expm1-undefine 51.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{\left(1 - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{2}\right) \cdot 0.5}{1 - \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}}\right)} - 1} \]

8. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\right)} - 1} \]

9. Taylor expanded in x around -inf 5.5%

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

    1. metadata-eval 5.5%

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

11. Applied egg-rr 5.5%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Developer Target 1: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024172 
(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))))))))