Given's Rotation SVD example

Percentage Accurate: 79.0% → 99.9%

Time: 10.1s

Alternatives: 7

Speedup: 0.7×

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: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

   1. Initial program 20.1%

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

      1. add-log-exp 20.1%

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

      2. +-commutative 20.1%

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

      3. distribute-rgt-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. fma-define 20.1%

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

   4. Applied egg-rr 20.1%

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

   5. Taylor expanded in x around -inf 45.0%

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

      1. mul-1-neg 45.0%

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

      2. distribute-rgt-out 45.0%

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

      3. metadata-eval 45.0%

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

   7. Simplified 45.0%

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

   8. Taylor expanded in p around 0 53.9%

      \[\leadsto -\color{blue}{p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 53.9%

         \[\leadsto -p \cdot \color{blue}{\left(\frac{1}{x} + -1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)} \]

      2. distribute-rgt-in 53.8%

         \[\leadsto -\color{blue}{\left(\frac{1}{x} \cdot p + \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot p\right)} \]

      3. associate-*l/ 54.0%

         \[\leadsto -\left(\color{blue}{\frac{1 \cdot p}{x}} + \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot p\right) \]

      4. *-lft-identity 54.0%

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

      5. associate-*r/ 54.0%

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

      6. associate-*l/ 54.0%

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

      7. associate-*r* 54.0%

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

      8. unpow2 54.0%

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

      9. unpow3 54.0%

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

      10. associate-*r/ 54.0%

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

      11. cube-div 55.5%

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

   10. Simplified 55.5%

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

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

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 50.8%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 88.0%

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

Alternative 2: 69.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 6.9 \cdot 10^{-169}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\log e\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 6.9e-169) (/ p_m (- x)) (if (<= p_m 1.3e-36) (log E) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 6.9e-169) {
		tmp = p_m / -x;
	} else if (p_m <= 1.3e-36) {
		tmp = log(((double) M_E));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 6.9e-169) {
		tmp = p_m / -x;
	} else if (p_m <= 1.3e-36) {
		tmp = Math.log(Math.E);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 6.9e-169:
		tmp = p_m / -x
	elif p_m <= 1.3e-36:
		tmp = math.log(math.e)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 6.9e-169)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 1.3e-36)
		tmp = log(exp(1));
	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 <= 6.9e-169)
		tmp = p_m / -x;
	elseif (p_m <= 1.3e-36)
		tmp = log(2.71828182845904523536);
	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, 6.9e-169], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 1.3e-36], N[Log[E], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 6.90000000000000027e-169

   1. Initial program 73.9%

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

      1. add-log-exp 73.9%

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

      2. +-commutative 73.9%

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

      3. distribute-rgt-in 73.9%

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

      4. metadata-eval 73.9%

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

      5. fma-define 73.9%

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in x around -inf 17.5%

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

      1. associate-*r/ 17.5%

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

      2. neg-mul-1 17.5%

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

   7. Simplified 17.5%

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

   if 6.90000000000000027e-169 < p < 1.3e-36

   1. Initial program 72.2%

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

      1. add-log-exp 72.1%

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

      2. +-commutative 72.1%

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

      3. distribute-rgt-in 72.1%

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

      4. metadata-eval 72.1%

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

      5. fma-define 72.1%

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

   4. Applied egg-rr 72.1%

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

   5. Taylor expanded in x around inf 53.1%

      \[\leadsto \log \color{blue}{\left(e^{1}\right)} \]
   6. Step-by-step derivation

      1. exp-1-e 53.1%

         \[\leadsto \log \color{blue}{e} \]

   7. Simplified 53.1%

      \[\leadsto \log \color{blue}{e} \]

   if 1.3e-36 < p

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0 87.1%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6.9 \cdot 10^{-169}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\log e\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\log e\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -3.2e-114) (/ p_m (- x)) (log E)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.2e-114) {
		tmp = p_m / -x;
	} else {
		tmp = log(((double) M_E));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -3.2e-114) {
		tmp = p_m / -x;
	} else {
		tmp = Math.log(Math.E);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.2e-114:
		tmp = p_m / -x
	else:
		tmp = math.log(math.e)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.2e-114)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = log(exp(1));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -3.2e-114)
		tmp = p_m / -x;
	else
		tmp = log(2.71828182845904523536);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.2e-114], N[(p$95$m / (-x)), $MachinePrecision], N[Log[E], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002e-114

   1. Initial program 51.2%

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

      1. add-log-exp 51.2%

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

      2. +-commutative 51.2%

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

      3. distribute-rgt-in 51.2%

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

      4. metadata-eval 51.2%

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

      5. fma-define 51.2%

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

   4. Applied egg-rr 51.2%

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

   5. Taylor expanded in x around -inf 36.1%

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

      1. associate-*r/ 36.1%

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

      2. neg-mul-1 36.1%

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

   7. Simplified 36.1%

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

   if -3.2000000000000002e-114 < x

   1. Initial program 99.3%

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

      1. add-log-exp 99.3%

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

      2. +-commutative 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. fma-define 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around inf 51.9%

      \[\leadsto \log \color{blue}{\left(e^{1}\right)} \]
   6. Step-by-step derivation

      1. exp-1-e 51.9%

         \[\leadsto \log \color{blue}{e} \]

   7. Simplified 51.9%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\log e\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 35.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2.2e-131) (/ p_m (- x)) 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.2e-131) {
		tmp = p_m / -x;
	} else {
		tmp = 1.5;
	}
	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 (x <= (-2.2d-131)) then
        tmp = p_m / -x
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -2.2e-131) {
		tmp = p_m / -x;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.2e-131:
		tmp = p_m / -x
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.2e-131)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -2.2e-131)
		tmp = p_m / -x;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.2e-131], N[(p$95$m / (-x)), $MachinePrecision], 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -2.2e-131

   1. Initial program 53.3%

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

      1. add-log-exp 53.2%

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

      2. +-commutative 53.2%

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

      3. distribute-rgt-in 53.2%

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

      4. metadata-eval 53.2%

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

      5. fma-define 53.2%

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

   4. Applied egg-rr 53.2%

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

   5. Taylor expanded in x around -inf 34.1%

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

      1. associate-*r/ 34.1%

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

      2. neg-mul-1 34.1%

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

   7. Simplified 34.1%

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

   if -2.2e-131 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.9%

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

   5. Applied egg-rr 19.3%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 17.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -5e+20) 0.0 1.5))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e+20) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	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 (x <= (-5d+20)) then
        tmp = 0.0d0
    else
        tmp = 1.5d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -5e+20) {
		tmp = 0.0;
	} else {
		tmp = 1.5;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e+20:
		tmp = 0.0
	else:
		tmp = 1.5
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e+20)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e+20)
		tmp = 0.0;
	else
		tmp = 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e+20], 0.0, 1.5]

Derivation

1. Split input into 2 regimes

2. if x < -5e20

   1. Initial program 53.6%

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

   3. Taylor expanded in x around -inf 22.5%

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

      1. neg-mul-1 22.5%

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

   5. Simplified 22.5%

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

   6. Taylor expanded in x around 0 22.5%

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

   if -5e20 < x

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0 61.0%

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

   5. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 15.6% accurate, 35.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+20}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -5e+20) 0.0 0.125))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e+20) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	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 <= (-5d+20)) then
        tmp = 0.0d0
    else
        tmp = 0.125d0
    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 <= -5e+20) {
		tmp = 0.0;
	} else {
		tmp = 0.125;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e+20:
		tmp = 0.0
	else:
		tmp = 0.125
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e+20)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e+20)
		tmp = 0.0;
	else
		tmp = 0.125;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e+20], 0.0, 0.125]

Derivation

1. Split input into 2 regimes

2. if x < -5e20

   1. Initial program 53.6%

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

   3. Taylor expanded in x around -inf 22.5%

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

      1. neg-mul-1 22.5%

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

   5. Simplified 22.5%

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

   6. Taylor expanded in x around 0 22.5%

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

   if -5e20 < x

   1. Initial program 85.0%

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

   3. Taylor expanded in x around 0 61.0%

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

   5. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{0.125} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 6.6% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Taylor expanded in x around -inf 7.4%

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

   1. neg-mul-1 7.4%

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

5. Simplified 7.4%

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

6. Taylor expanded in x around 0 7.4%

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

Developer Target 1: 79.0% 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 2024180 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

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