Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.7%

Time: 15.0s

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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.99998:\\ \;\;\;\;\frac{p\_m + \frac{{p\_m}^{3}}{{x}^{2}} \cdot -1.5}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.99998)
   (/ (+ p_m (* (/ (pow p_m 3.0) (pow x 2.0)) -1.5)) (- x))
   (sqrt (* 0.5 (log (exp (+ 1.0 (/ x (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)))) <= -0.99998) {
		tmp = (p_m + ((pow(p_m, 3.0) / pow(x, 2.0)) * -1.5)) / -x;
	} else {
		tmp = sqrt((0.5 * log(exp((1.0 + (x / 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)))) <= -0.99998) {
		tmp = (p_m + ((Math.pow(p_m, 3.0) / Math.pow(x, 2.0)) * -1.5)) / -x;
	} else {
		tmp = Math.sqrt((0.5 * Math.log(Math.exp((1.0 + (x / 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)))) <= -0.99998:
		tmp = (p_m + ((math.pow(p_m, 3.0) / math.pow(x, 2.0)) * -1.5)) / -x
	else:
		tmp = math.sqrt((0.5 * math.log(math.exp((1.0 + (x / 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)))) <= -0.99998)
		tmp = Float64(Float64(p_m + Float64(Float64((p_m ^ 3.0) / (x ^ 2.0)) * -1.5)) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * log(exp(Float64(1.0 + Float64(x / 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)))) <= -0.99998)
		tmp = (p_m + (((p_m ^ 3.0) / (x ^ 2.0)) * -1.5)) / -x;
	else
		tmp = sqrt((0.5 * log(exp((1.0 + (x / 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], -0.99998], N[(N[(p$95$m + N[(N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99997999999999998

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

      2. sqr-neg 21.2%

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

      3. associate-*l* 21.2%

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

      4. sqr-neg 21.2%

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

      5. fma-define 21.2%

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

      6. sqr-neg 21.2%

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

      7. fma-define 21.2%

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

      8. associate-*l* 21.2%

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

      9. +-commutative 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in x around -inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. distribute-rgt-out 61.8%

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

      3. metadata-eval 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in x around inf 66.4%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 66.4%

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

   10. Simplified 66.4%

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

   if -0.99997999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. fma-undefine 99.9%

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

      5. fma-undefine 99.9%

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

      6. associate-*r* 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. hypot-define 99.9%

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

      9. associate-*r* 99.9%

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

      10. *-commutative 99.9%

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

      11. sqrt-prod 99.9%

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

      12. sqrt-prod 47.6%

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

      13. add-sqr-sqrt 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99998:\\ \;\;\;\;\frac{p + \frac{{p}^{3}}{{x}^{2}} \cdot -1.5}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.99998:\\ \;\;\;\;\frac{p\_m + \frac{{p\_m}^{3}}{{x}^{2}} \cdot -1.5}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(x, \frac{1}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.99998)
   (/ (+ p_m (* (/ (pow p_m 3.0) (pow x 2.0)) -1.5)) (- x))
   (sqrt (* 0.5 (fma x (/ 1.0 (hypot x (* p_m 2.0))) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.99998) {
		tmp = (p_m + ((pow(p_m, 3.0) / pow(x, 2.0)) * -1.5)) / -x;
	} else {
		tmp = sqrt((0.5 * fma(x, (1.0 / hypot(x, (p_m * 2.0))), 1.0)));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.99998)
		tmp = Float64(Float64(p_m + Float64(Float64((p_m ^ 3.0) / (x ^ 2.0)) * -1.5)) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * fma(x, Float64(1.0 / hypot(x, Float64(p_m * 2.0))), 1.0)));
	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], -0.99998], N[(N[(p$95$m + N[(N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(x * N[(1.0 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99997999999999998

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

      2. sqr-neg 21.2%

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

      3. associate-*l* 21.2%

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

      4. sqr-neg 21.2%

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

      5. fma-define 21.2%

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

      6. sqr-neg 21.2%

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

      7. fma-define 21.2%

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

      8. associate-*l* 21.2%

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

      9. +-commutative 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in x around -inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. distribute-rgt-out 61.8%

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

      3. metadata-eval 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in x around inf 66.4%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 66.4%

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

   10. Simplified 66.4%

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

   if -0.99997999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. fma-undefine 99.9%

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

      6. fma-define 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99998:\\ \;\;\;\;\frac{p + \frac{{p}^{3}}{{x}^{2}} \cdot -1.5}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(x, \frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.99998:\\ \;\;\;\;\frac{p\_m + \frac{{p\_m}^{3}}{{x}^{2}} \cdot -1.5}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.99998)
   (/ (+ p_m (* (/ (pow p_m 3.0) (pow x 2.0)) -1.5)) (- x))
   (sqrt (* 0.5 (+ 1.0 (/ x (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)))) <= -0.99998) {
		tmp = (p_m + ((pow(p_m, 3.0) / pow(x, 2.0)) * -1.5)) / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / 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)))) <= -0.99998) {
		tmp = (p_m + ((Math.pow(p_m, 3.0) / Math.pow(x, 2.0)) * -1.5)) / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / 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)))) <= -0.99998:
		tmp = (p_m + ((math.pow(p_m, 3.0) / math.pow(x, 2.0)) * -1.5)) / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / 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)))) <= -0.99998)
		tmp = Float64(Float64(p_m + Float64(Float64((p_m ^ 3.0) / (x ^ 2.0)) * -1.5)) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / 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)))) <= -0.99998)
		tmp = (p_m + (((p_m ^ 3.0) / (x ^ 2.0)) * -1.5)) / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / 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], -0.99998], N[(N[(p$95$m + N[(N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99997999999999998

   1. Initial program 21.2%

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

      1. +-commutative 21.2%

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

      2. sqr-neg 21.2%

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

      3. associate-*l* 21.2%

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

      4. sqr-neg 21.2%

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

      5. fma-define 21.2%

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

      6. sqr-neg 21.2%

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

      7. fma-define 21.2%

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

      8. associate-*l* 21.2%

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

      9. +-commutative 21.2%

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

   3. Simplified 21.2%

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

   5. Taylor expanded in x around -inf 61.8%

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

      1. mul-1-neg 61.8%

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

      2. distribute-rgt-out 61.8%

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

      3. metadata-eval 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in x around inf 66.4%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 66.4%

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

   10. Simplified 66.4%

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

   if -0.99997999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. sqr-neg 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqr-neg 99.9%

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

      5. fma-define 99.9%

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

      6. sqr-neg 99.9%

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

      7. fma-define 99.9%

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

      8. associate-*l* 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-undefine 99.9%

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

      3. associate-*r* 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt1-in 99.9%

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

      6. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99998:\\ \;\;\;\;\frac{p + \frac{{p}^{3}}{{x}^{2}} \cdot -1.5}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1.2e+30)
   (/ p_m (- x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot x (* p_m 2.0))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.2e+30) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / 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 <= -1.2e+30) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.2e+30:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot(x, (p_m * 2.0))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.2e+30)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / 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 <= -1.2e+30)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / 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, -1.2e+30], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e30

   1. Initial program 45.6%

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

      1. +-commutative 45.6%

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

      2. sqr-neg 45.6%

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

      3. associate-*l* 45.6%

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

      4. sqr-neg 45.6%

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

      5. fma-define 45.6%

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

      6. sqr-neg 45.6%

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

      7. fma-define 45.6%

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

      8. associate-*l* 45.6%

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

      9. +-commutative 45.6%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in x around -inf 54.2%

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

      1. mul-1-neg 54.2%

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

      2. distribute-neg-frac2 54.2%

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

   7. Simplified 54.2%

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

   if -1.2e30 < x

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. sqr-neg 88.6%

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

      3. associate-*l* 88.6%

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

      4. sqr-neg 88.6%

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

      5. fma-define 88.6%

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

      6. sqr-neg 88.6%

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

      7. fma-define 88.6%

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

      8. associate-*l* 88.6%

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

      9. +-commutative 88.6%

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

   3. Simplified 88.6%

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

      1. *-commutative 88.6%

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

      2. fma-undefine 88.6%

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

      3. associate-*r* 88.6%

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

      4. +-commutative 88.6%

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

      5. distribute-rgt1-in 88.6%

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

      6. +-commutative 88.6%

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

   6. Applied egg-rr 88.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p\_m} \cdot \frac{x}{p\_m}\right)}\\ \mathbf{elif}\;p\_m \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{p\_m \cdot \left(\frac{1.5}{x} + \frac{x}{{p\_m}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.4e-224)
   (/ p_m (- x))
   (if (<= p_m 5.5e-78)
     (+ 1.0 (/ 1.0 (* -2.0 (* (/ x p_m) (/ x p_m)))))
     (if (<= p_m 3e-15)
       (/ -1.0 (* p_m (+ (/ 1.5 x) (/ x (pow p_m 2.0)))))
       (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.4e-224) {
		tmp = p_m / -x;
	} else if (p_m <= 5.5e-78) {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	} else if (p_m <= 3e-15) {
		tmp = -1.0 / (p_m * ((1.5 / x) + (x / pow(p_m, 2.0))));
	} 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 <= 2.4d-224) then
        tmp = p_m / -x
    else if (p_m <= 5.5d-78) then
        tmp = 1.0d0 + (1.0d0 / ((-2.0d0) * ((x / p_m) * (x / p_m))))
    else if (p_m <= 3d-15) then
        tmp = (-1.0d0) / (p_m * ((1.5d0 / x) + (x / (p_m ** 2.0d0))))
    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 <= 2.4e-224) {
		tmp = p_m / -x;
	} else if (p_m <= 5.5e-78) {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	} else if (p_m <= 3e-15) {
		tmp = -1.0 / (p_m * ((1.5 / x) + (x / Math.pow(p_m, 2.0))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.4e-224:
		tmp = p_m / -x
	elif p_m <= 5.5e-78:
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))))
	elif p_m <= 3e-15:
		tmp = -1.0 / (p_m * ((1.5 / x) + (x / math.pow(p_m, 2.0))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.4e-224)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 5.5e-78)
		tmp = Float64(1.0 + Float64(1.0 / Float64(-2.0 * Float64(Float64(x / p_m) * Float64(x / p_m)))));
	elseif (p_m <= 3e-15)
		tmp = Float64(-1.0 / Float64(p_m * Float64(Float64(1.5 / x) + Float64(x / (p_m ^ 2.0)))));
	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 <= 2.4e-224)
		tmp = p_m / -x;
	elseif (p_m <= 5.5e-78)
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	elseif (p_m <= 3e-15)
		tmp = -1.0 / (p_m * ((1.5 / x) + (x / (p_m ^ 2.0))));
	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, 2.4e-224], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 5.5e-78], N[(1.0 + N[(1.0 / N[(-2.0 * N[(N[(x / p$95$m), $MachinePrecision] * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 3e-15], N[(-1.0 / N[(p$95$m * N[(N[(1.5 / x), $MachinePrecision] + N[(x / N[Power[p$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if p < 2.40000000000000014e-224

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-*l* 80.7%

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

      4. sqr-neg 80.7%

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

      5. fma-define 80.7%

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

      6. sqr-neg 80.7%

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

      7. fma-define 80.7%

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

      8. associate-*l* 80.7%

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

      9. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in x around -inf 14.6%

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

      1. mul-1-neg 14.6%

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

      2. distribute-neg-frac2 14.6%

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

   7. Simplified 14.6%

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

   if 2.40000000000000014e-224 < p < 5.50000000000000017e-78

   1. Initial program 63.4%

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

      1. +-commutative 63.4%

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

      2. sqr-neg 63.4%

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

      3. associate-*l* 63.4%

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

      4. sqr-neg 63.4%

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

      5. fma-define 63.4%

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

      6. sqr-neg 63.4%

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

      7. fma-define 63.4%

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

      8. associate-*l* 63.4%

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

      9. +-commutative 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around inf 59.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{1 + \frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. clear-num 59.7%

         \[\leadsto 1 + \color{blue}{\frac{1}{\frac{{x}^{2}}{-0.5 \cdot {p}^{2}}}} \]

      2. inv-pow 59.7%

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

      3. *-un-lft-identity 59.7%

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

      4. times-frac 59.7%

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

      5. metadata-eval 59.7%

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

      6. unpow2 59.7%

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

      7. unpow2 59.7%

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

      8. frac-times 59.7%

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

      9. pow2 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. unpow-1 59.7%

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

   11. Simplified 59.7%

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

       1. unpow2 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 5.50000000000000017e-78 < p < 3e-15

   1. Initial program 33.4%

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

      1. +-commutative 33.4%

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

      2. sqr-neg 33.4%

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

      3. associate-*l* 33.4%

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

      4. sqr-neg 33.4%

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

      5. fma-define 33.4%

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

      6. sqr-neg 33.4%

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

      7. fma-define 33.4%

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

      8. associate-*l* 33.4%

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

      9. +-commutative 33.4%

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

   3. Simplified 33.4%

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

   5. Taylor expanded in x around -inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. distribute-rgt-out 69.6%

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

      3. metadata-eval 69.6%

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

   7. Simplified 69.6%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]
   8. Step-by-step derivation

      1. clear-num 69.4%

         \[\leadsto -\color{blue}{\frac{1}{\frac{x}{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}}} \]

      2. inv-pow 69.4%

         \[\leadsto -\color{blue}{{\left(\frac{x}{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}\right)}^{-1}} \]

      3. +-commutative 69.4%

         \[\leadsto -{\left(\frac{x}{\color{blue}{0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}} + p}}\right)}^{-1} \]

      4. *-commutative 69.4%

         \[\leadsto -{\left(\frac{x}{\color{blue}{\frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}} \cdot 0.125} + p}\right)}^{-1} \]

      5. fma-define 69.4%

         \[\leadsto -{\left(\frac{x}{\color{blue}{\mathsf{fma}\left(\frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}, 0.125, p\right)}}\right)}^{-1} \]

      6. times-frac 69.4%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left(\color{blue}{\frac{{p}^{4}}{p} \cdot \frac{-12}{{x}^{2}}}, 0.125, p\right)}\right)}^{-1} \]

      7. pow1 69.4%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left(\frac{{p}^{4}}{\color{blue}{{p}^{1}}} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1} \]

      8. pow-div 69.4%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left(\color{blue}{{p}^{\left(4 - 1\right)}} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1} \]

      9. metadata-eval 69.4%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left({p}^{\color{blue}{3}} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1} \]

   9. Applied egg-rr 69.4%

      \[\leadsto -\color{blue}{{\left(\frac{x}{\mathsf{fma}\left({p}^{3} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 69.4%

          \[\leadsto -\color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left({p}^{3} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}}} \]

   11. Simplified 69.4%

       \[\leadsto -\color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left({p}^{3} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}}} \]

   12. Taylor expanded in x around inf 70.9%

       \[\leadsto -\frac{1}{\color{blue}{x \cdot \left(1.5 \cdot \frac{p}{{x}^{2}} + \frac{1}{p}\right)}} \]

   13. Taylor expanded in p around inf 70.6%

       \[\leadsto -\frac{1}{\color{blue}{p \cdot \left(1.5 \cdot \frac{1}{x} + \frac{x}{{p}^{2}}\right)}} \]
   14. Step-by-step derivation

       1. associate-*r/ 70.6%

          \[\leadsto -\frac{1}{p \cdot \left(\color{blue}{\frac{1.5 \cdot 1}{x}} + \frac{x}{{p}^{2}}\right)} \]

       2. metadata-eval 70.6%

          \[\leadsto -\frac{1}{p \cdot \left(\frac{\color{blue}{1.5}}{x} + \frac{x}{{p}^{2}}\right)} \]

   15. Simplified 70.6%

       \[\leadsto -\frac{1}{\color{blue}{p \cdot \left(\frac{1.5}{x} + \frac{x}{{p}^{2}}\right)}} \]

   if 3e-15 < p

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0 82.2%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p} \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq 3 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{p \cdot \left(\frac{1.5}{x} + \frac{x}{{p}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.15 \cdot 10^{-224}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p\_m} \cdot \frac{x}{p\_m}\right)}\\ \mathbf{elif}\;p\_m \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\frac{x + 1.5 \cdot \frac{{p\_m}^{2}}{x}}{p\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.15e-224)
   (/ p_m (- x))
   (if (<= p_m 1.05e-78)
     (+ 1.0 (/ 1.0 (* -2.0 (* (/ x p_m) (/ x p_m)))))
     (if (<= p_m 2.6e-6)
       (/ -1.0 (/ (+ x (* 1.5 (/ (pow p_m 2.0) x))) p_m))
       (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.15e-224) {
		tmp = p_m / -x;
	} else if (p_m <= 1.05e-78) {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	} else if (p_m <= 2.6e-6) {
		tmp = -1.0 / ((x + (1.5 * (pow(p_m, 2.0) / 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 <= 2.15d-224) then
        tmp = p_m / -x
    else if (p_m <= 1.05d-78) then
        tmp = 1.0d0 + (1.0d0 / ((-2.0d0) * ((x / p_m) * (x / p_m))))
    else if (p_m <= 2.6d-6) then
        tmp = (-1.0d0) / ((x + (1.5d0 * ((p_m ** 2.0d0) / 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 <= 2.15e-224) {
		tmp = p_m / -x;
	} else if (p_m <= 1.05e-78) {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	} else if (p_m <= 2.6e-6) {
		tmp = -1.0 / ((x + (1.5 * (Math.pow(p_m, 2.0) / 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 <= 2.15e-224:
		tmp = p_m / -x
	elif p_m <= 1.05e-78:
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))))
	elif p_m <= 2.6e-6:
		tmp = -1.0 / ((x + (1.5 * (math.pow(p_m, 2.0) / 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 <= 2.15e-224)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 1.05e-78)
		tmp = Float64(1.0 + Float64(1.0 / Float64(-2.0 * Float64(Float64(x / p_m) * Float64(x / p_m)))));
	elseif (p_m <= 2.6e-6)
		tmp = Float64(-1.0 / Float64(Float64(x + Float64(1.5 * Float64((p_m ^ 2.0) / 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 <= 2.15e-224)
		tmp = p_m / -x;
	elseif (p_m <= 1.05e-78)
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	elseif (p_m <= 2.6e-6)
		tmp = -1.0 / ((x + (1.5 * ((p_m ^ 2.0) / 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, 2.15e-224], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 1.05e-78], N[(1.0 + N[(1.0 / N[(-2.0 * N[(N[(x / p$95$m), $MachinePrecision] * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 2.6e-6], N[(-1.0 / N[(N[(x + N[(1.5 * N[(N[Power[p$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if p < 2.15e-224

   1. Initial program 80.7%

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

      1. +-commutative 80.7%

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

      2. sqr-neg 80.7%

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

      3. associate-*l* 80.7%

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

      4. sqr-neg 80.7%

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

      5. fma-define 80.7%

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

      6. sqr-neg 80.7%

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

      7. fma-define 80.7%

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

      8. associate-*l* 80.7%

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

      9. +-commutative 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in x around -inf 14.6%

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

      1. mul-1-neg 14.6%

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

      2. distribute-neg-frac2 14.6%

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

   7. Simplified 14.6%

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

   if 2.15e-224 < p < 1.05e-78

   1. Initial program 63.4%

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

      1. +-commutative 63.4%

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

      2. sqr-neg 63.4%

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

      3. associate-*l* 63.4%

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

      4. sqr-neg 63.4%

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

      5. fma-define 63.4%

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

      6. sqr-neg 63.4%

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

      7. fma-define 63.4%

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

      8. associate-*l* 63.4%

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

      9. +-commutative 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around inf 59.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{1 + \frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. clear-num 59.7%

         \[\leadsto 1 + \color{blue}{\frac{1}{\frac{{x}^{2}}{-0.5 \cdot {p}^{2}}}} \]

      2. inv-pow 59.7%

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

      3. *-un-lft-identity 59.7%

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

      4. times-frac 59.7%

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

      5. metadata-eval 59.7%

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

      6. unpow2 59.7%

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

      7. unpow2 59.7%

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

      8. frac-times 59.7%

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

      9. pow2 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. unpow-1 59.7%

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

   11. Simplified 59.7%

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

       1. unpow2 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 1.05e-78 < p < 2.60000000000000009e-6

   1. Initial program 31.3%

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

      1. +-commutative 31.3%

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

      2. sqr-neg 31.3%

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

      3. associate-*l* 31.3%

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

      4. sqr-neg 31.3%

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

      5. fma-define 31.3%

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

      6. sqr-neg 31.3%

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

      7. fma-define 31.3%

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

      8. associate-*l* 31.3%

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

      9. +-commutative 31.3%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in x around -inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. distribute-rgt-out 71.8%

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

      3. metadata-eval 71.8%

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

   7. Simplified 71.8%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]
   8. Step-by-step derivation

      1. clear-num 71.6%

         \[\leadsto -\color{blue}{\frac{1}{\frac{x}{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}}} \]

      2. inv-pow 71.6%

         \[\leadsto -\color{blue}{{\left(\frac{x}{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}\right)}^{-1}} \]

      3. +-commutative 71.6%

         \[\leadsto -{\left(\frac{x}{\color{blue}{0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}} + p}}\right)}^{-1} \]

      4. *-commutative 71.6%

         \[\leadsto -{\left(\frac{x}{\color{blue}{\frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}} \cdot 0.125} + p}\right)}^{-1} \]

      5. fma-define 71.6%

         \[\leadsto -{\left(\frac{x}{\color{blue}{\mathsf{fma}\left(\frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}, 0.125, p\right)}}\right)}^{-1} \]

      6. times-frac 71.6%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left(\color{blue}{\frac{{p}^{4}}{p} \cdot \frac{-12}{{x}^{2}}}, 0.125, p\right)}\right)}^{-1} \]

      7. pow1 71.6%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left(\frac{{p}^{4}}{\color{blue}{{p}^{1}}} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1} \]

      8. pow-div 71.6%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left(\color{blue}{{p}^{\left(4 - 1\right)}} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1} \]

      9. metadata-eval 71.6%

         \[\leadsto -{\left(\frac{x}{\mathsf{fma}\left({p}^{\color{blue}{3}} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1} \]

   9. Applied egg-rr 71.6%

      \[\leadsto -\color{blue}{{\left(\frac{x}{\mathsf{fma}\left({p}^{3} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 71.6%

          \[\leadsto -\color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left({p}^{3} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}}} \]

   11. Simplified 71.6%

       \[\leadsto -\color{blue}{\frac{1}{\frac{x}{\mathsf{fma}\left({p}^{3} \cdot \frac{-12}{{x}^{2}}, 0.125, p\right)}}} \]

   12. Taylor expanded in p around 0 73.2%

       \[\leadsto -\frac{1}{\color{blue}{\frac{x + 1.5 \cdot \frac{{p}^{2}}{x}}{p}}} \]

   if 2.60000000000000009e-6 < p

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0 83.4%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.15 \cdot 10^{-224}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p} \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{-1}{\frac{x + 1.5 \cdot \frac{{p}^{2}}{x}}{p}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 2 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p\_m} \cdot \frac{x}{p\_m}\right)}\\ \mathbf{elif}\;p\_m \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 2e-224)
     t_0
     (if (<= p_m 1.25e-79)
       (+ 1.0 (/ 1.0 (* -2.0 (* (/ x p_m) (/ x p_m)))))
       (if (<= p_m 2.6e-6) t_0 (sqrt 0.5))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 2e-224) {
		tmp = t_0;
	} else if (p_m <= 1.25e-79) {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	} else if (p_m <= 2.6e-6) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 2d-224) then
        tmp = t_0
    else if (p_m <= 1.25d-79) then
        tmp = 1.0d0 + (1.0d0 / ((-2.0d0) * ((x / p_m) * (x / p_m))))
    else if (p_m <= 2.6d-6) then
        tmp = t_0
    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 t_0 = p_m / -x;
	double tmp;
	if (p_m <= 2e-224) {
		tmp = t_0;
	} else if (p_m <= 1.25e-79) {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	} else if (p_m <= 2.6e-6) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 2e-224:
		tmp = t_0
	elif p_m <= 1.25e-79:
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))))
	elif p_m <= 2.6e-6:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 2e-224)
		tmp = t_0;
	elseif (p_m <= 1.25e-79)
		tmp = Float64(1.0 + Float64(1.0 / Float64(-2.0 * Float64(Float64(x / p_m) * Float64(x / p_m)))));
	elseif (p_m <= 2.6e-6)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 2e-224)
		tmp = t_0;
	elseif (p_m <= 1.25e-79)
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	elseif (p_m <= 2.6e-6)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 2e-224], t$95$0, If[LessEqual[p$95$m, 1.25e-79], N[(1.0 + N[(1.0 / N[(-2.0 * N[(N[(x / p$95$m), $MachinePrecision] * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 2.6e-6], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2e-224 or 1.25e-79 < p < 2.60000000000000009e-6

   1. Initial program 76.5%

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

      1. +-commutative 76.5%

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

      2. sqr-neg 76.5%

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

      3. associate-*l* 76.5%

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

      4. sqr-neg 76.5%

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

      5. fma-define 76.5%

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

      6. sqr-neg 76.5%

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

      7. fma-define 76.5%

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

      8. associate-*l* 76.5%

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

      9. +-commutative 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in x around -inf 19.7%

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

      1. mul-1-neg 19.7%

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

      2. distribute-neg-frac2 19.7%

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

   7. Simplified 19.7%

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

   if 2e-224 < p < 1.25e-79

   1. Initial program 63.4%

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

      1. +-commutative 63.4%

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

      2. sqr-neg 63.4%

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

      3. associate-*l* 63.4%

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

      4. sqr-neg 63.4%

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

      5. fma-define 63.4%

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

      6. sqr-neg 63.4%

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

      7. fma-define 63.4%

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

      8. associate-*l* 63.4%

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

      9. +-commutative 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around inf 59.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 59.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]

   7. Simplified 59.7%

      \[\leadsto \color{blue}{1 + \frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. clear-num 59.7%

         \[\leadsto 1 + \color{blue}{\frac{1}{\frac{{x}^{2}}{-0.5 \cdot {p}^{2}}}} \]

      2. inv-pow 59.7%

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

      3. *-un-lft-identity 59.7%

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

      4. times-frac 59.7%

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

      5. metadata-eval 59.7%

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

      6. unpow2 59.7%

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

      7. unpow2 59.7%

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

      8. frac-times 59.7%

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

      9. pow2 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. unpow-1 59.7%

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

   11. Simplified 59.7%

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

       1. unpow2 59.7%

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

   13. Applied egg-rr 59.7%

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

   if 2.60000000000000009e-6 < p

   1. Initial program 89.9%

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

   3. Taylor expanded in x around 0 83.4%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-79}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p} \cdot \frac{x}{p}\right)}\\ \mathbf{elif}\;p \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p\_m} \cdot \frac{x}{p\_m}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x 8.2e-275)
   (/ p_m (- x))
   (+ 1.0 (/ 1.0 (* -2.0 (* (/ x p_m) (/ x p_m)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= 8.2e-275) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.2d-275) then
        tmp = p_m / -x
    else
        tmp = 1.0d0 + (1.0d0 / ((-2.0d0) * ((x / p_m) * (x / p_m))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= 8.2e-275) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= 8.2e-275:
		tmp = p_m / -x
	else:
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= 8.2e-275)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(1.0 + Float64(1.0 / Float64(-2.0 * Float64(Float64(x / p_m) * Float64(x / p_m)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= 8.2e-275)
		tmp = p_m / -x;
	else
		tmp = 1.0 + (1.0 / (-2.0 * ((x / p_m) * (x / p_m))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, 8.2e-275], N[(p$95$m / (-x)), $MachinePrecision], N[(1.0 + N[(1.0 / N[(-2.0 * N[(N[(x / p$95$m), $MachinePrecision] * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.19999999999999949e-275

   1. Initial program 59.9%

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

      1. +-commutative 59.9%

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

      2. sqr-neg 59.9%

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

      3. associate-*l* 59.9%

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

      4. sqr-neg 59.9%

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

      5. fma-define 59.9%

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

      6. sqr-neg 59.9%

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

      7. fma-define 59.9%

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

      8. associate-*l* 59.9%

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

      9. +-commutative 59.9%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in x around -inf 35.3%

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

      1. mul-1-neg 35.3%

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

      2. distribute-neg-frac2 35.3%

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

   7. Simplified 35.3%

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

   if 8.19999999999999949e-275 < 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. sqr-neg 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. sqr-neg 100.0%

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

      7. fma-define 100.0%

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

      8. associate-*l* 100.0%

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

      9. +-commutative 100.0%

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

   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. Taylor expanded in x around inf 49.5%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 49.5%

         \[\leadsto 1 + \color{blue}{\frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]

   7. Simplified 49.5%

      \[\leadsto \color{blue}{1 + \frac{-0.5 \cdot {p}^{2}}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. clear-num 49.5%

         \[\leadsto 1 + \color{blue}{\frac{1}{\frac{{x}^{2}}{-0.5 \cdot {p}^{2}}}} \]

      2. inv-pow 49.5%

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

      3. *-un-lft-identity 49.5%

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

      4. times-frac 49.5%

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

      5. metadata-eval 49.5%

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

      6. unpow2 49.5%

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

      7. unpow2 49.5%

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

      8. frac-times 49.5%

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

      9. pow2 49.5%

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

   9. Applied egg-rr 49.5%

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

       1. unpow-1 49.5%

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

   11. Simplified 49.5%

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

       1. unpow2 49.5%

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

   13. Applied egg-rr 49.5%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{-2 \cdot \left(\frac{x}{p} \cdot \frac{x}{p}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 27.0% accurate, 53.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \frac{p\_m}{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

p_m = abs(p)
function code(p_m, x)
	return Float64(p_m / Float64(-x))
end

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[(p$95$m / (-x)), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. +-commutative 78.7%

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

   2. sqr-neg 78.7%

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

   3. associate-*l* 78.7%

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

   4. sqr-neg 78.7%

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

   5. fma-define 78.7%

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

   6. sqr-neg 78.7%

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

   7. fma-define 78.7%

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

   8. associate-*l* 78.7%

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

   9. +-commutative 78.7%

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

3. Simplified 78.7%

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

5. Taylor expanded in x around -inf 20.4%

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

   1. mul-1-neg 20.4%

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

   2. distribute-neg-frac2 20.4%

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

7. Simplified 20.4%

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

8. Final simplification 20.4%

   \[\leadsto \frac{p}{-x} \]
9. Add Preprocessing

Developer target: 79.1% 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 2024082 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))