Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.8%

Time: 13.0s

Alternatives: 5

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: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.99:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.99)
   (/ p_m (- x))
   (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)))) <= -0.99) {
		tmp = p_m / -x;
	} 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)))) <= -0.99) {
		tmp = p_m / -x;
	} 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)))) <= -0.99:
		tmp = p_m / -x
	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)))) <= -0.99)
		tmp = Float64(p_m / Float64(-x));
	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)))) <= -0.99)
		tmp = p_m / -x;
	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], -0.99], N[(p$95$m / (-x)), $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)))) < -0.98999999999999999

   1. Initial program 16.1%

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

      1. add-sqr-sqrt 16.1%

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

      2. hypot-define 16.1%

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

      3. associate-*l* 16.1%

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

      4. sqrt-prod 16.1%

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

      5. metadata-eval 16.1%

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

      6. sqrt-unprod 4.8%

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

      7. add-sqr-sqrt 16.1%

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

   4. Applied egg-rr 16.1%

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

      1. sqrt-prod 16.1%

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

      2. add-sqr-sqrt 0.0%

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

      3. metadata-eval 0.0%

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

      4. add-sqr-sqrt 16.1%

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

      5. hypot-undefine 16.1%

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

      6. +-commutative 16.1%

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

      7. hypot-undefine 16.1%

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

      8. associate-+r+ 16.0%

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

      9. *-un-lft-identity 16.0%

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

      10. *-un-lft-identity 16.0%

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

      11. sqrt-prod 16.0%

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

      12. add-cbrt-cube 16.0%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

   6. Applied egg-rr 16.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

   7. Taylor expanded in x around -inf 54.4%

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

      1. mul-1-neg 54.4%

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

      2. distribute-neg-frac2 54.4%

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

   9. Simplified 54.4%

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

   if -0.98999999999999999 < (/.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-sqr-sqrt 99.9%

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

      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 55.7%

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

      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 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 1.6 \cdot 10^{-237}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.2 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 5.7 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 1.6e-237)
     1.0
     (if (<= p_m 2.2e-198)
       t_0
       (if (<= p_m 4.5e-161) 1.0 (if (<= p_m 5.7e-39) 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 <= 1.6e-237) {
		tmp = 1.0;
	} else if (p_m <= 2.2e-198) {
		tmp = t_0;
	} else if (p_m <= 4.5e-161) {
		tmp = 1.0;
	} else if (p_m <= 5.7e-39) {
		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 <= 1.6d-237) then
        tmp = 1.0d0
    else if (p_m <= 2.2d-198) then
        tmp = t_0
    else if (p_m <= 4.5d-161) then
        tmp = 1.0d0
    else if (p_m <= 5.7d-39) 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 <= 1.6e-237) {
		tmp = 1.0;
	} else if (p_m <= 2.2e-198) {
		tmp = t_0;
	} else if (p_m <= 4.5e-161) {
		tmp = 1.0;
	} else if (p_m <= 5.7e-39) {
		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 <= 1.6e-237:
		tmp = 1.0
	elif p_m <= 2.2e-198:
		tmp = t_0
	elif p_m <= 4.5e-161:
		tmp = 1.0
	elif p_m <= 5.7e-39:
		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 <= 1.6e-237)
		tmp = 1.0;
	elseif (p_m <= 2.2e-198)
		tmp = t_0;
	elseif (p_m <= 4.5e-161)
		tmp = 1.0;
	elseif (p_m <= 5.7e-39)
		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 <= 1.6e-237)
		tmp = 1.0;
	elseif (p_m <= 2.2e-198)
		tmp = t_0;
	elseif (p_m <= 4.5e-161)
		tmp = 1.0;
	elseif (p_m <= 5.7e-39)
		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, 1.6e-237], 1.0, If[LessEqual[p$95$m, 2.2e-198], t$95$0, If[LessEqual[p$95$m, 4.5e-161], 1.0, If[LessEqual[p$95$m, 5.7e-39], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.6e-237 or 2.2e-198 < p < 4.4999999999999996e-161

   1. Initial program 73.0%

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

      1. add-sqr-sqrt 73.0%

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

      2. hypot-define 73.0%

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

      3. associate-*l* 73.0%

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

      4. sqrt-prod 73.0%

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

      5. metadata-eval 73.0%

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

      6. sqrt-unprod 10.1%

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

      7. add-sqr-sqrt 73.0%

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

   4. Applied egg-rr 73.0%

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

      1. sqrt-prod 72.5%

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

      2. add-sqr-sqrt 51.3%

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

      3. metadata-eval 51.3%

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

      4. add-sqr-sqrt 72.5%

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

      5. hypot-undefine 72.5%

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

      6. +-commutative 72.5%

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

      7. hypot-undefine 72.5%

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

      8. associate-+r+ 72.5%

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

      9. *-un-lft-identity 72.5%

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

      10. *-un-lft-identity 72.5%

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

      11. sqrt-prod 73.0%

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

      12. add-cbrt-cube 73.0%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

   6. Applied egg-rr 73.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

   7. Taylor expanded in x around inf 39.8%

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

   if 1.6e-237 < p < 2.2e-198 or 4.4999999999999996e-161 < p < 5.6999999999999997e-39

   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-sqr-sqrt 53.3%

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

      2. hypot-define 53.3%

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

      3. associate-*l* 53.3%

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

      4. sqrt-prod 53.3%

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

      5. metadata-eval 53.3%

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

      6. sqrt-unprod 53.3%

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

      7. add-sqr-sqrt 53.3%

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

   4. Applied egg-rr 53.3%

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

      1. sqrt-prod 52.8%

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

      2. add-sqr-sqrt 38.9%

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

      3. metadata-eval 38.9%

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

      4. add-sqr-sqrt 52.8%

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

      5. hypot-undefine 52.8%

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

      6. +-commutative 52.8%

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

      7. hypot-undefine 52.8%

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

      8. associate-+r+ 52.8%

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

      9. *-un-lft-identity 52.8%

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

      10. *-un-lft-identity 52.8%

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

      11. sqrt-prod 53.3%

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

      12. add-cbrt-cube 53.4%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

   6. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

   7. Taylor expanded in x around -inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. distribute-neg-frac2 54.1%

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

   9. Simplified 54.1%

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

   if 5.6999999999999997e-39 < p

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0 88.7%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.6 \cdot 10^{-237}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 4.5 \cdot 10^{-161}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-194}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2.3e-194) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.3e-194) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.3d-194)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -2.3e-194) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.3e-194:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.3e-194)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -2.3e-194)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.3e-194], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.30000000000000003e-194

   1. Initial program 54.1%

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

      1. add-sqr-sqrt 54.1%

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

      2. hypot-define 54.1%

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

      3. associate-*l* 54.1%

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

      4. sqrt-prod 54.1%

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

      5. metadata-eval 54.1%

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

      6. sqrt-unprod 29.5%

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

      7. add-sqr-sqrt 54.1%

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

   4. Applied egg-rr 54.1%

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

      1. sqrt-prod 54.1%

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

      2. add-sqr-sqrt 5.4%

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

      3. metadata-eval 5.4%

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

      4. add-sqr-sqrt 54.1%

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

      5. hypot-undefine 54.1%

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

      6. +-commutative 54.1%

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

      7. hypot-undefine 54.1%

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

      8. associate-+r+ 54.1%

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

      9. *-un-lft-identity 54.1%

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

      10. *-un-lft-identity 54.1%

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

      11. sqrt-prod 54.1%

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

      12. add-cbrt-cube 54.1%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

   6. Applied egg-rr 54.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

   7. Taylor expanded in x around -inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. distribute-neg-frac2 31.3%

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

   9. Simplified 31.3%

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

   if -2.30000000000000003e-194 < 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 54.0%

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

      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%

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

      1. sqrt-prod 99.2%

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

      2. add-sqr-sqrt 99.2%

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

      3. metadata-eval 99.2%

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

      4. add-sqr-sqrt 99.2%

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

      5. hypot-undefine 99.2%

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

      6. +-commutative 99.2%

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

      7. hypot-undefine 99.2%

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

      8. associate-+r+ 99.2%

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

      9. *-un-lft-identity 99.2%

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

      10. *-un-lft-identity 99.2%

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

      11. sqrt-prod 100.0%

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

      12. add-cbrt-cube 99.9%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

   7. Taylor expanded in x around inf 59.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-194}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.6% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.4e+82) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.4e+82) {
		tmp = p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.4d+82)) then
        tmp = p_m / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -3.4e+82) {
		tmp = p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.4e+82:
		tmp = p_m / x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.4e+82)
		tmp = Float64(p_m / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -3.4e+82)
		tmp = p_m / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.4e+82], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999994e82

   1. Initial program 50.4%

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

   3. Taylor expanded in x around -inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. associate-/l* 45.4%

         \[\leadsto -\color{blue}{p \cdot \frac{\sqrt{0.5} \cdot \sqrt{2}}{x}} \]

      3. distribute-rgt-neg-in 45.4%

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

      4. associate-/l* 45.5%

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

   5. Simplified 45.5%

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

      1. pow1 45.5%

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

      2. add-sqr-sqrt 45.5%

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

      3. sqrt-unprod 45.5%

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

      4. sqr-neg 45.5%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 58.6%

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

      7. associate-*r/ 58.3%

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

      8. sqrt-unprod 58.9%

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

      9. metadata-eval 58.9%

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

      10. metadata-eval 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. unpow1 58.9%

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

      2. associate-*r/ 59.0%

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

      3. *-rgt-identity 59.0%

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

   9. Simplified 59.0%

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

   if -3.39999999999999994e82 < x

   1. Initial program 80.2%

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

      1. add-sqr-sqrt 80.2%

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

      2. hypot-define 80.2%

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

      3. associate-*l* 80.2%

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

      4. sqrt-prod 80.2%

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

      5. metadata-eval 80.2%

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

      6. sqrt-unprod 44.1%

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

      7. add-sqr-sqrt 80.2%

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

   4. Applied egg-rr 80.2%

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

      1. sqrt-prod 79.8%

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

      2. add-sqr-sqrt 58.4%

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

      3. metadata-eval 58.4%

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

      4. add-sqr-sqrt 79.8%

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

      5. hypot-undefine 79.7%

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

      6. +-commutative 79.7%

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

      7. hypot-undefine 79.8%

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

      8. associate-+r+ 79.7%

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

      9. *-un-lft-identity 79.7%

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

      10. *-un-lft-identity 79.7%

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

      11. sqrt-prod 80.2%

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

      12. add-cbrt-cube 80.1%

          \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

   6. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

   7. Taylor expanded in x around inf 39.1%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+82}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.4% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. add-sqr-sqrt 76.7%

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

   2. hypot-define 76.7%

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

   3. associate-*l* 76.7%

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

   4. sqrt-prod 76.7%

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

   5. metadata-eval 76.7%

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

   6. sqrt-unprod 41.5%

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

   7. add-sqr-sqrt 76.7%

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

4. Applied egg-rr 76.7%

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

   1. sqrt-prod 76.3%

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

   2. add-sqr-sqrt 51.6%

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

   3. metadata-eval 51.6%

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

   4. add-sqr-sqrt 76.3%

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

   5. hypot-undefine 76.3%

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

   6. +-commutative 76.3%

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

   7. hypot-undefine 76.3%

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

   8. associate-+r+ 76.3%

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

   9. *-un-lft-identity 76.3%

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

   10. *-un-lft-identity 76.3%

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

   11. sqrt-prod 76.7%

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

   12. add-cbrt-cube 76.7%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}\right) \cdot \sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)\right)}}} \]

6. Applied egg-rr 76.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5\right)}^{1.5}}} \]

7. Taylor expanded in x around inf 35.4%

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

8. Final simplification 35.4%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024075 
(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))))))))