Given's Rotation SVD example

Percentage Accurate: 79.0% → 99.8%

Time: 7.4s

Alternatives: 8

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: 99.8% accurate, 0.7× speedup

Math

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

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (/ (- p) x)
   (sqrt (+ 0.5 (* (/ x (hypot x (* p 2.0))) 0.5)))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 + ((x / hypot(x, (p * 2.0))) * 0.5)));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 + ((x / Math.hypot(x, (p * 2.0))) * 0.5)));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 + ((x / math.hypot(x, (p * 2.0))) * 0.5)))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x / hypot(x, Float64(p * 2.0))) * 0.5)));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -1.0)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 + ((x / hypot(x, (p * 2.0))) * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

   1. Initial program 14.8%

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

      1. +-commutative 14.8%

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

      2. distribute-rgt-in 14.8%

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

      3. +-commutative 14.8%

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

      4. add-sqr-sqrt 14.8%

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

      5. hypot-def 14.8%

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

      6. associate-*l* 14.8%

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

      7. sqrt-prod 14.8%

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

      8. metadata-eval 14.8%

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

      9. sqrt-unprod 4.5%

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

      10. add-sqr-sqrt 14.8%

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

      11. metadata-eval 14.8%

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

   3. Applied egg-rr 14.8%

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

   4. Taylor expanded in x around -inf 50.3%

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

      1. neg-mul-1 50.3%

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

      2. distribute-neg-frac 50.3%

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

   6. Simplified 50.3%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-def 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 52.4%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 86.2%

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

Alternative 2: 77.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+49}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.2:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= x -3.8e+49)
     t_0
     (if (<= x -1.2)
       (sqrt 0.5)
       (if (<= x -1.5e-9)
         t_0
         (if (<= x -2.3e-82)
           (sqrt (+ 0.5 (* 0.25 (/ x p))))
           (if (<= x -3.2e-137)
             t_0
             (sqrt (* 0.5 (+ 1.0 (/ x (+ x (* 2.0 (/ (* p p) x))))))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (x <= -3.8e+49) {
		tmp = t_0;
	} else if (x <= -1.2) {
		tmp = sqrt(0.5);
	} else if (x <= -1.5e-9) {
		tmp = t_0;
	} else if (x <= -2.3e-82) {
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	} else if (x <= -3.2e-137) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (x <= (-3.8d+49)) then
        tmp = t_0
    else if (x <= (-1.2d0)) then
        tmp = sqrt(0.5d0)
    else if (x <= (-1.5d-9)) then
        tmp = t_0
    else if (x <= (-2.3d-82)) then
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p))))
    else if (x <= (-3.2d-137)) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (x + (2.0d0 * ((p * p) / x)))))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (x <= -3.8e+49) {
		tmp = t_0;
	} else if (x <= -1.2) {
		tmp = Math.sqrt(0.5);
	} else if (x <= -1.5e-9) {
		tmp = t_0;
	} else if (x <= -2.3e-82) {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p))));
	} else if (x <= -3.2e-137) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if x <= -3.8e+49:
		tmp = t_0
	elif x <= -1.2:
		tmp = math.sqrt(0.5)
	elif x <= -1.5e-9:
		tmp = t_0
	elif x <= -2.3e-82:
		tmp = math.sqrt((0.5 + (0.25 * (x / p))))
	elif x <= -3.2e-137:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (x <= -3.8e+49)
		tmp = t_0;
	elseif (x <= -1.2)
		tmp = sqrt(0.5);
	elseif (x <= -1.5e-9)
		tmp = t_0;
	elseif (x <= -2.3e-82)
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p))));
	elseif (x <= -3.2e-137)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(x + Float64(2.0 * Float64(Float64(p * p) / x)))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (x <= -3.8e+49)
		tmp = t_0;
	elseif (x <= -1.2)
		tmp = sqrt(0.5);
	elseif (x <= -1.5e-9)
		tmp = t_0;
	elseif (x <= -2.3e-82)
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	elseif (x <= -3.2e-137)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[((-p) / x), $MachinePrecision]}, If[LessEqual[x, -3.8e+49], t$95$0, If[LessEqual[x, -1.2], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[x, -1.5e-9], t$95$0, If[LessEqual[x, -2.3e-82], N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, -3.2e-137], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(x + N[(2.0 * N[(N[(p * p), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.7999999999999999e49 or -1.19999999999999996 < x < -1.49999999999999999e-9 or -2.29999999999999997e-82 < x < -3.20000000000000021e-137

   1. Initial program 43.1%

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

      1. +-commutative 43.1%

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

      2. distribute-rgt-in 43.1%

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

      3. +-commutative 43.1%

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

      4. add-sqr-sqrt 43.1%

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

      5. hypot-def 43.1%

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

      6. associate-*l* 43.1%

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

      7. sqrt-prod 43.1%

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

      8. metadata-eval 43.1%

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

      9. sqrt-unprod 16.0%

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

      10. add-sqr-sqrt 43.1%

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

      11. metadata-eval 43.1%

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

   3. Applied egg-rr 43.1%

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

   4. Taylor expanded in x around -inf 36.1%

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

      1. neg-mul-1 36.1%

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

      2. distribute-neg-frac 36.1%

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

   6. Simplified 36.1%

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

   if -3.7999999999999999e49 < x < -1.19999999999999996

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 74.5%

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

   if -1.49999999999999999e-9 < x < -2.29999999999999997e-82

   1. Initial program 65.7%

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

   2. Taylor expanded in x around 0 63.6%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]

   if -3.20000000000000021e-137 < x

   1. Initial program 98.6%

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

   2. Taylor expanded in p around 0 98.3%

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

      1. unpow2 98.3%

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

   4. Simplified 98.3%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{x + 2 \cdot \frac{p \cdot p}{x}}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -1.2:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-82}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \]

Alternative 3: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 5 \cdot 10^{-180}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{elif}\;p \leq 6.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-5} \lor \neg \left(p \leq 1.05 \cdot 10^{+68}\right):\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 5e-180)
   (+ 1.0 (* -0.5 (/ (/ p x) (/ x p))))
   (if (<= p 6.2e-31)
     (/ (- p) x)
     (if (or (<= p 3.8e-5) (not (<= p 1.05e+68)))
       (sqrt (+ 0.5 (* 0.25 (/ x p))))
       1.0))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 5e-180) {
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	} else if (p <= 6.2e-31) {
		tmp = -p / x;
	} else if ((p <= 3.8e-5) || !(p <= 1.05e+68)) {
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 5d-180) then
        tmp = 1.0d0 + ((-0.5d0) * ((p / x) / (x / p)))
    else if (p <= 6.2d-31) then
        tmp = -p / x
    else if ((p <= 3.8d-5) .or. (.not. (p <= 1.05d+68))) then
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p))))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 5e-180) {
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	} else if (p <= 6.2e-31) {
		tmp = -p / x;
	} else if ((p <= 3.8e-5) || !(p <= 1.05e+68)) {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 5e-180:
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)))
	elif p <= 6.2e-31:
		tmp = -p / x
	elif (p <= 3.8e-5) or not (p <= 1.05e+68):
		tmp = math.sqrt((0.5 + (0.25 * (x / p))))
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 5e-180)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) / Float64(x / p))));
	elseif (p <= 6.2e-31)
		tmp = Float64(Float64(-p) / x);
	elseif ((p <= 3.8e-5) || !(p <= 1.05e+68))
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 5e-180)
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	elseif (p <= 6.2e-31)
		tmp = -p / x;
	elseif ((p <= 3.8e-5) || ~((p <= 1.05e+68)))
		tmp = sqrt((0.5 + (0.25 * (x / p))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 5e-180], N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] / N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 6.2e-31], N[((-p) / x), $MachinePrecision], If[Or[LessEqual[p, 3.8e-5], N[Not[LessEqual[p, 1.05e+68]], $MachinePrecision]], N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 4 regimes

2. if p < 5.0000000000000001e-180

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. distribute-rgt-in 74.6%

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

      3. +-commutative 74.6%

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

      4. add-sqr-sqrt 74.6%

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

      5. hypot-def 74.6%

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

      6. associate-*l* 74.6%

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

      7. sqrt-prod 74.6%

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

      8. metadata-eval 74.6%

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

      9. sqrt-unprod 13.1%

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

      10. add-sqr-sqrt 74.6%

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

      11. metadata-eval 74.6%

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

   3. Applied egg-rr 74.6%

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

   4. Taylor expanded in x around inf 34.5%

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

      1. unpow2 34.5%

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

      2. unpow2 34.5%

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

      3. times-frac 34.5%

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

      4. unpow2 34.5%

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

   6. Simplified 34.5%

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

      1. unpow2 34.5%

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

      2. clear-num 34.5%

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

      3. un-div-inv 34.5%

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

   8. Applied egg-rr 34.5%

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

   if 5.0000000000000001e-180 < p < 6.19999999999999999e-31

   1. Initial program 42.6%

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

      1. +-commutative 42.6%

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

      2. distribute-rgt-in 42.6%

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

      3. +-commutative 42.6%

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

      4. add-sqr-sqrt 42.6%

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

      5. hypot-def 42.6%

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

      6. associate-*l* 42.6%

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

      7. sqrt-prod 42.6%

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

      8. metadata-eval 42.6%

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

      9. sqrt-unprod 42.6%

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

      10. add-sqr-sqrt 42.6%

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

      11. metadata-eval 42.6%

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

   3. Applied egg-rr 42.6%

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

   4. Taylor expanded in x around -inf 61.4%

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

      1. neg-mul-1 61.4%

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

      2. distribute-neg-frac 61.4%

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

   6. Simplified 61.4%

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

   if 6.19999999999999999e-31 < p < 3.8000000000000002e-5 or 1.05e68 < p

   1. Initial program 96.8%

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

   2. Taylor expanded in x around 0 95.1%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]

   if 3.8000000000000002e-5 < p < 1.05e68

   1. Initial program 67.8%

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

      1. +-commutative 67.8%

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

      2. distribute-rgt-in 67.8%

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

      3. +-commutative 67.8%

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

      4. add-sqr-sqrt 67.8%

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

      5. hypot-def 67.8%

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

      6. associate-*l* 67.8%

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

      7. sqrt-prod 67.8%

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

      8. metadata-eval 67.8%

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

      9. sqrt-unprod 67.8%

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

      10. add-sqr-sqrt 67.8%

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

      11. metadata-eval 67.8%

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

   3. Applied egg-rr 67.8%

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

   4. Taylor expanded in x around inf 53.5%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5 \cdot 10^{-180}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{elif}\;p \leq 6.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-5} \lor \neg \left(p \leq 1.05 \cdot 10^{+68}\right):\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 66.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := 1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{if}\;p \leq 4.3 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 5.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.00048:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4.75 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (/ (/ p x) (/ x p))))))
   (if (<= p 4.3e-182)
     t_0
     (if (<= p 5.6e-35)
       (/ (- p) x)
       (if (<= p 0.00048) (sqrt 0.5) (if (<= p 4.75e+33) t_0 (sqrt 0.5)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = 1.0 + (-0.5 * ((p / x) / (x / p)));
	double tmp;
	if (p <= 4.3e-182) {
		tmp = t_0;
	} else if (p <= 5.6e-35) {
		tmp = -p / x;
	} else if (p <= 0.00048) {
		tmp = sqrt(0.5);
	} else if (p <= 4.75e+33) {
		tmp = t_0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.5d0) * ((p / x) / (x / p)))
    if (p <= 4.3d-182) then
        tmp = t_0
    else if (p <= 5.6d-35) then
        tmp = -p / x
    else if (p <= 0.00048d0) then
        tmp = sqrt(0.5d0)
    else if (p <= 4.75d+33) then
        tmp = t_0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = 1.0 + (-0.5 * ((p / x) / (x / p)));
	double tmp;
	if (p <= 4.3e-182) {
		tmp = t_0;
	} else if (p <= 5.6e-35) {
		tmp = -p / x;
	} else if (p <= 0.00048) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 4.75e+33) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = 1.0 + (-0.5 * ((p / x) / (x / p)))
	tmp = 0
	if p <= 4.3e-182:
		tmp = t_0
	elif p <= 5.6e-35:
		tmp = -p / x
	elif p <= 0.00048:
		tmp = math.sqrt(0.5)
	elif p <= 4.75e+33:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) / Float64(x / p))))
	tmp = 0.0
	if (p <= 4.3e-182)
		tmp = t_0;
	elseif (p <= 5.6e-35)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 0.00048)
		tmp = sqrt(0.5);
	elseif (p <= 4.75e+33)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = 1.0 + (-0.5 * ((p / x) / (x / p)));
	tmp = 0.0;
	if (p <= 4.3e-182)
		tmp = t_0;
	elseif (p <= 5.6e-35)
		tmp = -p / x;
	elseif (p <= 0.00048)
		tmp = sqrt(0.5);
	elseif (p <= 4.75e+33)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] / N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, 4.3e-182], t$95$0, If[LessEqual[p, 5.6e-35], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 0.00048], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 4.75e+33], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.3e-182 or 4.80000000000000012e-4 < p < 4.7500000000000002e33

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. distribute-rgt-in 74.6%

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

      3. +-commutative 74.6%

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

      4. add-sqr-sqrt 74.6%

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

      5. hypot-def 74.6%

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

      6. associate-*l* 74.6%

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

      7. sqrt-prod 74.6%

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

      8. metadata-eval 74.6%

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

      9. sqrt-unprod 16.2%

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

      10. add-sqr-sqrt 74.6%

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

      11. metadata-eval 74.6%

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

   3. Applied egg-rr 74.6%

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

   4. Taylor expanded in x around inf 36.0%

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

      1. unpow2 36.0%

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

      2. unpow2 36.0%

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

      3. times-frac 36.0%

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

      4. unpow2 36.0%

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

   6. Simplified 36.0%

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

      1. unpow2 36.0%

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

      2. clear-num 36.0%

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

      3. un-div-inv 36.0%

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

   8. Applied egg-rr 36.0%

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

   if 4.3e-182 < p < 5.5999999999999999e-35

   1. Initial program 42.6%

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

      1. +-commutative 42.6%

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

      2. distribute-rgt-in 42.6%

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

      3. +-commutative 42.6%

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

      4. add-sqr-sqrt 42.6%

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

      5. hypot-def 42.6%

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

      6. associate-*l* 42.6%

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

      7. sqrt-prod 42.6%

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

      8. metadata-eval 42.6%

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

      9. sqrt-unprod 42.6%

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

      10. add-sqr-sqrt 42.6%

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

      11. metadata-eval 42.6%

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

   3. Applied egg-rr 42.6%

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

   4. Taylor expanded in x around -inf 61.4%

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

      1. neg-mul-1 61.4%

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

      2. distribute-neg-frac 61.4%

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

   6. Simplified 61.4%

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

   if 5.5999999999999999e-35 < p < 4.80000000000000012e-4 or 4.7500000000000002e33 < p

   1. Initial program 92.9%

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

   2. Taylor expanded in x around 0 88.9%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.3 \cdot 10^{-182}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{elif}\;p \leq 5.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.00048:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4.75 \cdot 10^{+33}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 66.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.135:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4.75 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 3.5e-180)
   (+ 1.0 (* -0.5 (/ (/ p x) (/ x p))))
   (if (<= p 2e-34)
     (/ (- p) x)
     (if (<= p 0.135) (sqrt 0.5) (if (<= p 4.75e+33) 1.0 (sqrt 0.5))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 3.5e-180) {
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	} else if (p <= 2e-34) {
		tmp = -p / x;
	} else if (p <= 0.135) {
		tmp = sqrt(0.5);
	} else if (p <= 4.75e+33) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 3.5d-180) then
        tmp = 1.0d0 + ((-0.5d0) * ((p / x) / (x / p)))
    else if (p <= 2d-34) then
        tmp = -p / x
    else if (p <= 0.135d0) then
        tmp = sqrt(0.5d0)
    else if (p <= 4.75d+33) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 3.5e-180) {
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	} else if (p <= 2e-34) {
		tmp = -p / x;
	} else if (p <= 0.135) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 4.75e+33) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 3.5e-180:
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)))
	elif p <= 2e-34:
		tmp = -p / x
	elif p <= 0.135:
		tmp = math.sqrt(0.5)
	elif p <= 4.75e+33:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 3.5e-180)
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) / Float64(x / p))));
	elseif (p <= 2e-34)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 0.135)
		tmp = sqrt(0.5);
	elseif (p <= 4.75e+33)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 3.5e-180)
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	elseif (p <= 2e-34)
		tmp = -p / x;
	elseif (p <= 0.135)
		tmp = sqrt(0.5);
	elseif (p <= 4.75e+33)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 3.5e-180], N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] / N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[p, 2e-34], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 0.135], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 4.75e+33], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 3.5000000000000001e-180

   1. Initial program 74.6%

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

      1. +-commutative 74.6%

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

      2. distribute-rgt-in 74.6%

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

      3. +-commutative 74.6%

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

      4. add-sqr-sqrt 74.6%

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

      5. hypot-def 74.6%

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

      6. associate-*l* 74.6%

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

      7. sqrt-prod 74.6%

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

      8. metadata-eval 74.6%

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

      9. sqrt-unprod 13.1%

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

      10. add-sqr-sqrt 74.6%

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

      11. metadata-eval 74.6%

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

   3. Applied egg-rr 74.6%

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

   4. Taylor expanded in x around inf 34.5%

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

      1. unpow2 34.5%

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

      2. unpow2 34.5%

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

      3. times-frac 34.5%

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

      4. unpow2 34.5%

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

   6. Simplified 34.5%

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

      1. unpow2 34.5%

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

      2. clear-num 34.5%

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

      3. un-div-inv 34.5%

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

   8. Applied egg-rr 34.5%

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

   if 3.5000000000000001e-180 < p < 1.99999999999999986e-34

   1. Initial program 42.6%

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

      1. +-commutative 42.6%

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

      2. distribute-rgt-in 42.6%

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

      3. +-commutative 42.6%

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

      4. add-sqr-sqrt 42.6%

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

      5. hypot-def 42.6%

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

      6. associate-*l* 42.6%

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

      7. sqrt-prod 42.6%

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

      8. metadata-eval 42.6%

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

      9. sqrt-unprod 42.6%

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

      10. add-sqr-sqrt 42.6%

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

      11. metadata-eval 42.6%

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

   3. Applied egg-rr 42.6%

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

   4. Taylor expanded in x around -inf 61.4%

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

      1. neg-mul-1 61.4%

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

      2. distribute-neg-frac 61.4%

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

   6. Simplified 61.4%

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

   if 1.99999999999999986e-34 < p < 0.13500000000000001 or 4.7500000000000002e33 < p

   1. Initial program 92.9%

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

   2. Taylor expanded in x around 0 88.9%

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

   if 0.13500000000000001 < p < 4.7500000000000002e33

   1. Initial program 75.9%

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

      1. +-commutative 75.9%

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

      2. distribute-rgt-in 75.9%

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

      3. +-commutative 75.9%

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

      4. add-sqr-sqrt 75.9%

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

      5. hypot-def 75.9%

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

      6. associate-*l* 75.9%

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

      7. sqrt-prod 75.9%

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

      8. metadata-eval 75.9%

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

      9. sqrt-unprod 75.9%

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

      10. add-sqr-sqrt 75.9%

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

      11. metadata-eval 75.9%

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

   3. Applied egg-rr 75.9%

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

   4. Taylor expanded in x around inf 67.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.5 \cdot 10^{-180}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 0.135:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4.75 \cdot 10^{+33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 51.5% accurate, 16.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-228}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= x 3.9e-228) (/ (- p) x) (+ 1.0 (* -0.5 (/ (/ p x) (/ x p))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= 3.9e-228) {
		tmp = -p / x;
	} else {
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.9d-228) then
        tmp = -p / x
    else
        tmp = 1.0d0 + ((-0.5d0) * ((p / x) / (x / p)))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= 3.9e-228) {
		tmp = -p / x;
	} else {
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= 3.9e-228:
		tmp = -p / x
	else:
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= 3.9e-228)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = Float64(1.0 + Float64(-0.5 * Float64(Float64(p / x) / Float64(x / p))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= 3.9e-228)
		tmp = -p / x;
	else
		tmp = 1.0 + (-0.5 * ((p / x) / (x / p)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, 3.9e-228], N[((-p) / x), $MachinePrecision], N[(1.0 + N[(-0.5 * N[(N[(p / x), $MachinePrecision] / N[(x / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.90000000000000029e-228

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. distribute-rgt-in 51.6%

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

      3. +-commutative 51.6%

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

      4. add-sqr-sqrt 51.6%

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

      5. hypot-def 51.6%

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

      6. associate-*l* 51.6%

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

      7. sqrt-prod 51.6%

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

      8. metadata-eval 51.6%

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

      9. sqrt-unprod 26.6%

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

      10. add-sqr-sqrt 51.6%

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

      11. metadata-eval 51.6%

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

   3. Applied egg-rr 51.6%

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

   4. Taylor expanded in x around -inf 30.1%

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

      1. neg-mul-1 30.1%

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

      2. distribute-neg-frac 30.1%

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

   6. Simplified 30.1%

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

   if 3.90000000000000029e-228 < 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. distribute-rgt-in 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-def 100.0%

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

      6. associate-*l* 100.0%

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

      7. sqrt-prod 100.0%

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

      8. metadata-eval 100.0%

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

      9. sqrt-unprod 51.1%

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

      10. add-sqr-sqrt 100.0%

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

      11. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around inf 48.8%

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

      1. unpow2 48.8%

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

      2. unpow2 48.8%

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

      3. times-frac 48.8%

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

      4. unpow2 48.8%

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

   6. Simplified 48.8%

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

      1. unpow2 48.8%

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

      2. clear-num 48.8%

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

      3. un-div-inv 48.8%

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

   8. Applied egg-rr 48.8%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\frac{\frac{p}{x}}{\frac{x}{p}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{-228}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot \frac{\frac{p}{x}}{\frac{x}{p}}\\ \end{array} \]

Alternative 7: 28.8% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -5e-310) (/ (- p) x) (/ p x)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p / x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -p / x
    else
        tmp = p / x
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p / x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -5e-310:
		tmp = -p / x
	else:
		tmp = p / x
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = Float64(p / x);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -p / x;
	else
		tmp = p / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -5e-310], N[((-p) / x), $MachinePrecision], N[(p / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 51.6%

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

      1. +-commutative 51.6%

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

      2. distribute-rgt-in 51.6%

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

      3. +-commutative 51.6%

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

      4. add-sqr-sqrt 51.6%

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

      5. hypot-def 51.6%

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

      6. associate-*l* 51.6%

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

      7. sqrt-prod 51.6%

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

      8. metadata-eval 51.6%

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

      9. sqrt-unprod 26.6%

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

      10. add-sqr-sqrt 51.6%

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

      11. metadata-eval 51.6%

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

   3. Applied egg-rr 51.6%

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

   4. Taylor expanded in x around -inf 30.1%

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

      1. neg-mul-1 30.1%

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

      2. distribute-neg-frac 30.1%

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

   6. Simplified 30.1%

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

   if -4.999999999999985e-310 < 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. Taylor expanded in x around -inf 4.3%

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

      1. unpow2 4.3%

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

      2. unpow2 4.3%

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

      3. times-frac 4.6%

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

   4. Simplified 4.6%

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

   5. Taylor expanded in p around 0 3.3%

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

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]

Alternative 8: 6.2% accurate, 71.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \frac{p}{x} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (/ p x))

C

p = abs(p);
double code(double p, double x) {
	return p / x;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = p / x
end function

Java

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

Python

p = abs(p)
def code(p, x):
	return p / x

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

2. Taylor expanded in x around -inf 18.2%

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

   1. unpow2 18.2%

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

   2. unpow2 18.2%

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

   3. times-frac 23.7%

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

4. Simplified 23.7%

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

5. Taylor expanded in p around 0 19.8%

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

6. Final simplification 19.8%

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

Developer target: 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 2023271 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (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))))))))