Given's Rotation SVD example

Percentage Accurate: 78.6% → 99.8%

Time: 13.2s

Alternatives: 9

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: 78.6% 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, 0.2× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\\ \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 + {t_0}^{9}}{\mathsf{fma}\left(t_0, t_0 + -1, 1\right) \cdot \left(1 + \left({t_0}^{6} - {t_0}^{3}\right)\right)}}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (hypot x (* p 2.0)))))
   (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9998)
     (/ (- p) x)
     (sqrt
      (*
       0.5
       (/
        (+ 1.0 (pow t_0 9.0))
        (*
         (fma t_0 (+ t_0 -1.0) 1.0)
         (+ 1.0 (- (pow t_0 6.0) (pow t_0 3.0))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = x / hypot(x, (p * 2.0));
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * ((1.0 + pow(t_0, 9.0)) / (fma(t_0, (t_0 + -1.0), 1.0) * (1.0 + (pow(t_0, 6.0) - pow(t_0, 3.0)))))));
	}
	return tmp;
}

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(x / hypot(x, Float64(p * 2.0)))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.9998)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(Float64(1.0 + (t_0 ^ 9.0)) / Float64(fma(t_0, Float64(t_0 + -1.0), 1.0) * Float64(1.0 + Float64((t_0 ^ 6.0) - (t_0 ^ 3.0)))))));
	end
	return tmp
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9998], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(1.0 + N[Power[t$95$0, 9.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(t$95$0 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 + N[(N[Power[t$95$0, 6.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)))) < -0.99980000000000002

   1. Initial program 15.0%

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

   2. Taylor expanded in x around -inf 58.5%

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

   3. Taylor expanded in p around -inf 67.8%

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

      1. associate-*r/ 67.8%

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

      2. mul-1-neg 67.8%

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

   5. Simplified 67.8%

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

   if -0.99980000000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. 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-def 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. metadata-eval 99.9%

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

      6. sqrt-unprod 51.0%

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

      7. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}\right)\right)}}} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}\right)\right)}} \]

      2. *-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}, \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + -1, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}\right)\right)}} \]

      3. +-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, \color{blue}{-1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}\right)\right)}} \]

      4. *-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, -1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}\right)\right)}} \]

      5. *-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, -1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{3}\right)\right)}} \]

   7. Simplified 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, -1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \frac{1 + {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{9}}{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}, \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + -1, 1\right) \cdot \left(1 + \left({\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{6} - {\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}\right)\right)}}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9998)
   (/ (- p) x)
   (pow
    (pow (sqrt (* 0.5 (+ 1.0 (/ x (hypot x (* p 2.0)))))) 3.0)
    0.3333333333333333)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998) {
		tmp = -p / x;
	} else {
		tmp = pow(pow(sqrt((0.5 * (1.0 + (x / hypot(x, (p * 2.0)))))), 3.0), 0.3333333333333333);
	}
	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)))) <= -0.9998) {
		tmp = -p / x;
	} else {
		tmp = Math.pow(Math.pow(Math.sqrt((0.5 * (1.0 + (x / Math.hypot(x, (p * 2.0)))))), 3.0), 0.3333333333333333);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998:
		tmp = -p / x
	else:
		tmp = math.pow(math.pow(math.sqrt((0.5 * (1.0 + (x / math.hypot(x, (p * 2.0)))))), 3.0), 0.3333333333333333)
	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)))) <= -0.9998)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = (sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(x, Float64(p * 2.0)))))) ^ 3.0) ^ 0.3333333333333333;
	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)))) <= -0.9998)
		tmp = -p / x;
	else
		tmp = (sqrt((0.5 * (1.0 + (x / hypot(x, (p * 2.0)))))) ^ 3.0) ^ 0.3333333333333333;
	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], -0.9998], N[((-p) / x), $MachinePrecision], N[Power[N[Power[N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99980000000000002

   1. Initial program 15.0%

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

   2. Taylor expanded in x around -inf 58.5%

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

   3. Taylor expanded in p around -inf 67.8%

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

      1. associate-*r/ 67.8%

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

      2. mul-1-neg 67.8%

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

   5. Simplified 67.8%

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

   if -0.99980000000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. sqrt-prod 99.4%

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

      2. +-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. fma-udef 99.4%

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

      5. fma-udef 99.4%

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

      6. *-commutative 99.4%

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

      7. +-commutative 99.4%

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

      8. add-sqr-sqrt 82.3%

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

   3. Applied egg-rr 71.0%

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

      1. hypot-1-def 71.0%

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

      2. rem-square-sqrt 99.4%

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

   5. Simplified 99.4%

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

      1. add-cbrt-cube 99.9%

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

      2. pow1/3 99.4%

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

      3. pow3 99.4%

         \[\leadsto {\color{blue}{\left({\left(\sqrt{0.5} \cdot \sqrt{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. sqrt-unprod 99.9%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 99.9%

         \[\leadsto {\left({\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{p \cdot 2}\right)}\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{{\left({\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9998)
   (/ (- p) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	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)))) <= -0.9998) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9998:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	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)))) <= -0.9998)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	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)))) <= -0.9998)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	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], -0.9998], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 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 4 p) p) (*.f64 x x)))) < -0.99980000000000002

   1. Initial program 15.0%

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

   2. Taylor expanded in x around -inf 58.5%

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

   3. Taylor expanded in p around -inf 67.8%

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

      1. associate-*r/ 67.8%

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

      2. mul-1-neg 67.8%

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

   5. Simplified 67.8%

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

   if -0.99980000000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. 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-def 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. metadata-eval 99.9%

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

      6. sqrt-unprod 51.0%

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

      7. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9998:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 4: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := p \cdot \sqrt{{x}^{-2}}\\ \mathbf{if}\;p \leq 9.5 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-113}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 200000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (* p (sqrt (pow x -2.0)))))
   (if (<= p 9.5e-285)
     1.0
     (if (<= p 1.25e-182)
       t_0
       (if (<= p 9e-113)
         1.0
         (if (<= p 200000000.0)
           t_0
           (sqrt (* 0.5 (+ 1.0 (/ x (* p 2.0)))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = p * sqrt(pow(x, -2.0));
	double tmp;
	if (p <= 9.5e-285) {
		tmp = 1.0;
	} else if (p <= 1.25e-182) {
		tmp = t_0;
	} else if (p <= 9e-113) {
		tmp = 1.0;
	} else if (p <= 200000000.0) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = p * sqrt((x ** (-2.0d0)))
    if (p <= 9.5d-285) then
        tmp = 1.0d0
    else if (p <= 1.25d-182) then
        tmp = t_0
    else if (p <= 9d-113) then
        tmp = 1.0d0
    else if (p <= 200000000.0d0) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (p * 2.0d0)))))
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double t_0 = p * Math.sqrt(Math.pow(x, -2.0));
	double tmp;
	if (p <= 9.5e-285) {
		tmp = 1.0;
	} else if (p <= 1.25e-182) {
		tmp = t_0;
	} else if (p <= 9e-113) {
		tmp = 1.0;
	} else if (p <= 200000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = p * math.sqrt(math.pow(x, -2.0))
	tmp = 0
	if p <= 9.5e-285:
		tmp = 1.0
	elif p <= 1.25e-182:
		tmp = t_0
	elif p <= 9e-113:
		tmp = 1.0
	elif p <= 200000000.0:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(p * sqrt((x ^ -2.0)))
	tmp = 0.0
	if (p <= 9.5e-285)
		tmp = 1.0;
	elseif (p <= 1.25e-182)
		tmp = t_0;
	elseif (p <= 9e-113)
		tmp = 1.0;
	elseif (p <= 200000000.0)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(p * 2.0)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = p * sqrt((x ^ -2.0));
	tmp = 0.0;
	if (p <= 9.5e-285)
		tmp = 1.0;
	elseif (p <= 1.25e-182)
		tmp = t_0;
	elseif (p <= 9e-113)
		tmp = 1.0;
	elseif (p <= 200000000.0)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := Block[{t$95$0 = N[(p * N[Sqrt[N[Power[x, -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p, 9.5e-285], 1.0, If[LessEqual[p, 1.25e-182], t$95$0, If[LessEqual[p, 9e-113], 1.0, If[LessEqual[p, 200000000.0], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(p * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 9.4999999999999997e-285 or 1.25000000000000006e-182 < p < 9.0000000000000002e-113

   1. Initial program 81.4%

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

   2. Taylor expanded in x around inf 42.3%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 9.4999999999999997e-285 < p < 1.25000000000000006e-182 or 9.0000000000000002e-113 < p < 2e8

   1. Initial program 41.2%

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

   2. Taylor expanded in x around -inf 35.4%

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

      1. pow1/2 35.4%

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

      2. associate-*r* 35.4%

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

      3. metadata-eval 35.4%

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

      4. *-un-lft-identity 35.4%

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

      5. div-inv 35.4%

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

      6. unpow-prod-down 44.5%

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

      7. pow1/2 44.5%

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

      8. unpow2 44.5%

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

      9. sqrt-prod 65.4%

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

      10. add-sqr-sqrt 65.8%

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

      11. pow-flip 65.8%

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

      12. metadata-eval 65.8%

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

   4. Applied egg-rr 65.8%

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

      1. unpow1/2 65.8%

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

   6. Simplified 65.8%

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

   if 2e8 < p

   1. Initial program 93.8%

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

   2. Taylor expanded in p around inf 89.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 9.5 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-182}:\\ \;\;\;\;p \cdot \sqrt{{x}^{-2}}\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-113}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 200000000:\\ \;\;\;\;p \cdot \sqrt{{x}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \]

Alternative 5: 66.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 8 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 1.75 \cdot 10^{-113}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 185000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 8e-285)
     1.0
     (if (<= p 1.25e-191)
       t_0
       (if (<= p 1.75e-113)
         1.0
         (if (<= p 185000000.0)
           t_0
           (sqrt (* 0.5 (+ 1.0 (/ x (* p 2.0)))))))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 8e-285) {
		tmp = 1.0;
	} else if (p <= 1.25e-191) {
		tmp = t_0;
	} else if (p <= 1.75e-113) {
		tmp = 1.0;
	} else if (p <= 185000000.0) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (p <= 8d-285) then
        tmp = 1.0d0
    else if (p <= 1.25d-191) then
        tmp = t_0
    else if (p <= 1.75d-113) then
        tmp = 1.0d0
    else if (p <= 185000000.0d0) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (p * 2.0d0)))))
    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 (p <= 8e-285) {
		tmp = 1.0;
	} else if (p <= 1.25e-191) {
		tmp = t_0;
	} else if (p <= 1.75e-113) {
		tmp = 1.0;
	} else if (p <= 185000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= 8e-285:
		tmp = 1.0
	elif p <= 1.25e-191:
		tmp = t_0
	elif p <= 1.75e-113:
		tmp = 1.0
	elif p <= 185000000.0:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (p * 2.0)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 8e-285)
		tmp = 1.0;
	elseif (p <= 1.25e-191)
		tmp = t_0;
	elseif (p <= 1.75e-113)
		tmp = 1.0;
	elseif (p <= 185000000.0)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(p * 2.0)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= 8e-285)
		tmp = 1.0;
	elseif (p <= 1.25e-191)
		tmp = t_0;
	elseif (p <= 1.75e-113)
		tmp = 1.0;
	elseif (p <= 185000000.0)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (p * 2.0)))));
	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[p, 8e-285], 1.0, If[LessEqual[p, 1.25e-191], t$95$0, If[LessEqual[p, 1.75e-113], 1.0, If[LessEqual[p, 185000000.0], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(p * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 8.00000000000000059e-285 or 1.25e-191 < p < 1.75000000000000014e-113

   1. Initial program 81.2%

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

   2. Taylor expanded in x around inf 42.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 8.00000000000000059e-285 < p < 1.25e-191 or 1.75000000000000014e-113 < p < 1.85e8

   1. Initial program 40.7%

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

   2. Taylor expanded in x around -inf 34.8%

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

   3. Taylor expanded in p around -inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. mul-1-neg 64.2%

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

   5. Simplified 64.2%

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

   if 1.85e8 < p

   1. Initial program 93.8%

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

   2. Taylor expanded in p around inf 89.6%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 8 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.25 \cdot 10^{-191}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.75 \cdot 10^{-113}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 185000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{p \cdot 2}\right)}\\ \end{array} \]

Alternative 6: 66.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq 6.4 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.6 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 8.6 \cdot 10^{-113}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 185000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p 6.4e-285)
     1.0
     (if (<= p 6.6e-193)
       t_0
       (if (<= p 8.6e-113) 1.0 (if (<= p 185000000.0) t_0 (sqrt 0.5)))))))

C

p = abs(p);
double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= 6.4e-285) {
		tmp = 1.0;
	} else if (p <= 6.6e-193) {
		tmp = t_0;
	} else if (p <= 8.6e-113) {
		tmp = 1.0;
	} else if (p <= 185000000.0) {
		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 = -p / x
    if (p <= 6.4d-285) then
        tmp = 1.0d0
    else if (p <= 6.6d-193) then
        tmp = t_0
    else if (p <= 8.6d-113) then
        tmp = 1.0d0
    else if (p <= 185000000.0d0) 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 = -p / x;
	double tmp;
	if (p <= 6.4e-285) {
		tmp = 1.0;
	} else if (p <= 6.6e-193) {
		tmp = t_0;
	} else if (p <= 8.6e-113) {
		tmp = 1.0;
	} else if (p <= 185000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= 6.4e-285:
		tmp = 1.0
	elif p <= 6.6e-193:
		tmp = t_0
	elif p <= 8.6e-113:
		tmp = 1.0
	elif p <= 185000000.0:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= 6.4e-285)
		tmp = 1.0;
	elseif (p <= 6.6e-193)
		tmp = t_0;
	elseif (p <= 8.6e-113)
		tmp = 1.0;
	elseif (p <= 185000000.0)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= 6.4e-285)
		tmp = 1.0;
	elseif (p <= 6.6e-193)
		tmp = t_0;
	elseif (p <= 8.6e-113)
		tmp = 1.0;
	elseif (p <= 185000000.0)
		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[((-p) / x), $MachinePrecision]}, If[LessEqual[p, 6.4e-285], 1.0, If[LessEqual[p, 6.6e-193], t$95$0, If[LessEqual[p, 8.6e-113], 1.0, If[LessEqual[p, 185000000.0], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 6.40000000000000032e-285 or 6.5999999999999998e-193 < p < 8.6000000000000001e-113

   1. Initial program 81.2%

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

   2. Taylor expanded in x around inf 42.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{2}} \]

   if 6.40000000000000032e-285 < p < 6.5999999999999998e-193 or 8.6000000000000001e-113 < p < 1.85e8

   1. Initial program 40.7%

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

   2. Taylor expanded in x around -inf 34.8%

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

   3. Taylor expanded in p around -inf 64.2%

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

      1. associate-*r/ 64.2%

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

      2. mul-1-neg 64.2%

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

   5. Simplified 64.2%

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

   if 1.85e8 < p

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 89.9%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6.4 \cdot 10^{-285}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 6.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 8.6 \cdot 10^{-113}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 185000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 7: 67.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 400000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= p 400000000.0) (/ (- p) x) (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 400000000.0) {
		tmp = -p / x;
	} 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 <= 400000000.0d0) then
        tmp = -p / x
    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 <= 400000000.0) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 400000000.0:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 400000000.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 400000000.0)
		tmp = -p / x;
	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, 400000000.0], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 4e8

   1. Initial program 70.9%

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

   2. Taylor expanded in x around -inf 22.8%

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

   3. Taylor expanded in p around -inf 25.4%

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

      1. associate-*r/ 25.4%

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

      2. mul-1-neg 25.4%

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

   5. Simplified 25.4%

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

   if 4e8 < p

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 89.9%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 400000000:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8: 29.2% 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 53.7%

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

   2. Taylor expanded in x around -inf 34.4%

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

   3. Taylor expanded in p around -inf 38.7%

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

      1. associate-*r/ 38.7%

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

      2. mul-1-neg 38.7%

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

   5. Simplified 38.7%

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

   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 4.9%

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

   3. Taylor expanded in p around 0 3.5%

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

4. Final simplification 21.3%

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

Alternative 9: 6.3% 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 76.7%

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

2. Taylor expanded in x around -inf 19.8%

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

3. Taylor expanded in p around 0 14.5%

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

4. Final simplification 14.5%

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

Developer target: 78.6% 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 2023318 
(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))))))))