Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.8%

Time: 12.3s

Alternatives: 10

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: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt[3]{{\left(\frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)}^{3}}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.9995)
   (/ p_m (- x))
   (sqrt (+ 0.5 (* 0.5 (cbrt (pow (/ x (hypot x (* p_m 2.0))) 3.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + (0.5 * cbrt(pow((x / hypot(x, (p_m * 2.0))), 3.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * Math.cbrt(Math.pow((x / Math.hypot(x, (p_m * 2.0))), 3.0)))));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.9995)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * cbrt((Float64(x / hypot(x, Float64(p_m * 2.0))) ^ 3.0)))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9995], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[Power[N[Power[N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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)))) < -0.99950000000000006

   1. Initial program 15.7%

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

   3. Taylor expanded in x around -inf 53.2%

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

   4. Taylor expanded in p around -inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. distribute-neg-frac 64.7%

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

   6. Simplified 64.7%

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

   if -0.99950000000000006 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. hypot-define 99.9%

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

      5. associate-*l* 99.9%

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

      6. sqrt-prod 99.9%

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

      7. metadata-eval 99.9%

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

      8. sqrt-unprod 45.8%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. add0 99.9%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} + 0} \]

      2. rem-log-exp 99.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}} + 0 \]

      3. distribute-lft-in 99.9%

         \[\leadsto \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} + 0 \]

      4. metadata-eval 99.9%

         \[\leadsto \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0 \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0} \]
   7. Step-by-step derivation

      1. add0 99.9%

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

      2. *-commutative 99.9%

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

   8. Simplified 99.9%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \sqrt[3]{{\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}^{3}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{\frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)} + 1}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.9995)
   (/ p_m (- x))
   (sqrt (* 0.5 (log (exp (+ (/ x (hypot x (* p_m 2.0))) 1.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * log(exp(((x / hypot(x, (p_m * 2.0))) + 1.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 * Math.log(Math.exp(((x / Math.hypot(x, (p_m * 2.0))) + 1.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 * math.log(math.exp(((x / math.hypot(x, (p_m * 2.0))) + 1.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.9995)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * log(exp(Float64(Float64(x / hypot(x, Float64(p_m * 2.0))) + 1.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 * log(exp(((x / hypot(x, (p_m * 2.0))) + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9995], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 1.0), $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)))) < -0.99950000000000006

   1. Initial program 15.7%

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

   3. Taylor expanded in x around -inf 53.2%

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

   4. Taylor expanded in p around -inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. distribute-neg-frac 64.7%

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

   6. Simplified 64.7%

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

   if -0.99950000000000006 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 99.9%

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

      2. +-commutative 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. hypot-define 99.9%

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

      5. associate-*l* 99.9%

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

      6. sqrt-prod 99.9%

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

      7. metadata-eval 99.9%

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

      8. sqrt-unprod 45.8%

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

      9. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(x, \frac{1}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.9995)
   (/ p_m (- x))
   (sqrt (* 0.5 (fma x (/ 1.0 (hypot x (* p_m 2.0))) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * fma(x, (1.0 / hypot(x, (p_m * 2.0))), 1.0)));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.9995)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * fma(x, Float64(1.0 / hypot(x, Float64(p_m * 2.0))), 1.0)));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9995], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(x * N[(1.0 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99950000000000006

   1. Initial program 15.7%

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

   3. Taylor expanded in x around -inf 53.2%

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

   4. Taylor expanded in p around -inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. distribute-neg-frac 64.7%

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

   6. Simplified 64.7%

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

   if -0.99950000000000006 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 99.9%

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

      2. div-inv 99.9%

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

      3. fma-define 99.9%

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

      4. +-commutative 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. hypot-define 99.9%

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

      7. associate-*l* 99.9%

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

      8. sqrt-prod 99.9%

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

      9. metadata-eval 99.9%

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

      10. sqrt-unprod 45.8%

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

      11. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(x, \frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.9995)
   (/ p_m (- x))
   (sqrt (+ 0.5 (* 0.5 (/ x (hypot x (* p_m 2.0))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (x / Math.hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + (0.5 * (x / math.hypot(x, (p_m * 2.0))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.9995)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(x / hypot(x, Float64(p_m * 2.0))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.9995)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p_m * 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9995], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99950000000000006

   1. Initial program 15.7%

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

   3. Taylor expanded in x around -inf 53.2%

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

   4. Taylor expanded in p around -inf 64.7%

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

      1. neg-mul-1 64.7%

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

      2. distribute-neg-frac 64.7%

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

   6. Simplified 64.7%

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

   if -0.99950000000000006 < (/.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. Add Preprocessing
   3. Step-by-step derivation

      1. add0 99.9%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} + 0} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\sqrt{0.5 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5} + 0} \]
   5. Step-by-step derivation

      1. add0 99.9%

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

      2. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9995:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 7.8 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.46 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 6 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 7.8e-160)
     t_0
     (if (<= p_m 1.55e-109)
       1.0
       (if (<= p_m 1.46e-59)
         t_0
         (if (<= p_m 6e-39) 1.0 (sqrt (+ 0.5 (* 0.25 (/ x p_m))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 7.8e-160) {
		tmp = t_0;
	} else if (p_m <= 1.55e-109) {
		tmp = 1.0;
	} else if (p_m <= 1.46e-59) {
		tmp = t_0;
	} else if (p_m <= 6e-39) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5 + (0.25 * (x / p_m))));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 7.8d-160) then
        tmp = t_0
    else if (p_m <= 1.55d-109) then
        tmp = 1.0d0
    else if (p_m <= 1.46d-59) then
        tmp = t_0
    else if (p_m <= 6d-39) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p_m))))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 7.8e-160) {
		tmp = t_0;
	} else if (p_m <= 1.55e-109) {
		tmp = 1.0;
	} else if (p_m <= 1.46e-59) {
		tmp = t_0;
	} else if (p_m <= 6e-39) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p_m))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 7.8e-160:
		tmp = t_0
	elif p_m <= 1.55e-109:
		tmp = 1.0
	elif p_m <= 1.46e-59:
		tmp = t_0
	elif p_m <= 6e-39:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5 + (0.25 * (x / p_m))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 7.8e-160)
		tmp = t_0;
	elseif (p_m <= 1.55e-109)
		tmp = 1.0;
	elseif (p_m <= 1.46e-59)
		tmp = t_0;
	elseif (p_m <= 6e-39)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p_m))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 7.8e-160)
		tmp = t_0;
	elseif (p_m <= 1.55e-109)
		tmp = 1.0;
	elseif (p_m <= 1.46e-59)
		tmp = t_0;
	elseif (p_m <= 6e-39)
		tmp = 1.0;
	else
		tmp = sqrt((0.5 + (0.25 * (x / p_m))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 7.8e-160], t$95$0, If[LessEqual[p$95$m, 1.55e-109], 1.0, If[LessEqual[p$95$m, 1.46e-59], t$95$0, If[LessEqual[p$95$m, 6e-39], 1.0, N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.79999999999999979e-160 or 1.55e-109 < p < 1.45999999999999994e-59

   1. Initial program 72.3%

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

   3. Taylor expanded in x around -inf 17.4%

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

   4. Taylor expanded in p around -inf 21.0%

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

      1. neg-mul-1 21.0%

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

      2. distribute-neg-frac 21.0%

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

   6. Simplified 21.0%

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

   if 7.79999999999999979e-160 < p < 1.55e-109 or 1.45999999999999994e-59 < p < 6.00000000000000055e-39

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 74.6%

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

      2. +-commutative 74.6%

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

      3. add-sqr-sqrt 74.6%

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

      4. hypot-define 74.6%

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

      5. associate-*l* 74.6%

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

      6. sqrt-prod 74.6%

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

      7. metadata-eval 74.6%

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

      8. sqrt-unprod 74.6%

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

      9. add-sqr-sqrt 74.6%

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

   4. Applied egg-rr 74.6%

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

      1. add0 74.6%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} + 0} \]

      2. rem-log-exp 74.6%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}} + 0 \]

      3. distribute-lft-in 74.6%

         \[\leadsto \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} + 0 \]

      4. metadata-eval 74.6%

         \[\leadsto \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0 \]

   6. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0} \]
   7. Step-by-step derivation

      1. add0 74.6%

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

      2. *-commutative 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in x around inf 74.7%

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

   if 6.00000000000000055e-39 < p

   1. Initial program 80.9%

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

      1. add-log-exp 80.9%

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

      2. +-commutative 80.9%

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

      3. add-sqr-sqrt 80.9%

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

      4. hypot-define 80.9%

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

      5. associate-*l* 80.9%

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

      6. sqrt-prod 80.9%

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

      7. metadata-eval 80.9%

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

      8. sqrt-unprod 80.9%

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

      9. add-sqr-sqrt 80.9%

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

   4. Applied egg-rr 80.9%

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

      1. add0 80.9%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} + 0} \]

      2. rem-log-exp 80.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}} + 0 \]

      3. distribute-lft-in 80.9%

         \[\leadsto \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} + 0 \]

      4. metadata-eval 80.9%

         \[\leadsto \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0 \]

   6. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0} \]
   7. Step-by-step derivation

      1. add0 80.9%

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

      2. *-commutative 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in x around 0 76.1%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 7.8 \cdot 10^{-160}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.46 \cdot 10^{-59}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 6 \cdot 10^{-39}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 1.07 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.65 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 2 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 1.07e-159)
     t_0
     (if (<= p_m 1.65e-109)
       1.0
       (if (<= p_m 2.65e-59) t_0 (if (<= p_m 2e-42) 1.0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 1.07e-159) {
		tmp = t_0;
	} else if (p_m <= 1.65e-109) {
		tmp = 1.0;
	} else if (p_m <= 2.65e-59) {
		tmp = t_0;
	} else if (p_m <= 2e-42) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 1.07d-159) then
        tmp = t_0
    else if (p_m <= 1.65d-109) then
        tmp = 1.0d0
    else if (p_m <= 2.65d-59) then
        tmp = t_0
    else if (p_m <= 2d-42) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 1.07e-159) {
		tmp = t_0;
	} else if (p_m <= 1.65e-109) {
		tmp = 1.0;
	} else if (p_m <= 2.65e-59) {
		tmp = t_0;
	} else if (p_m <= 2e-42) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 1.07e-159:
		tmp = t_0
	elif p_m <= 1.65e-109:
		tmp = 1.0
	elif p_m <= 2.65e-59:
		tmp = t_0
	elif p_m <= 2e-42:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 1.07e-159)
		tmp = t_0;
	elseif (p_m <= 1.65e-109)
		tmp = 1.0;
	elseif (p_m <= 2.65e-59)
		tmp = t_0;
	elseif (p_m <= 2e-42)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 1.07e-159)
		tmp = t_0;
	elseif (p_m <= 1.65e-109)
		tmp = 1.0;
	elseif (p_m <= 2.65e-59)
		tmp = t_0;
	elseif (p_m <= 2e-42)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 1.07e-159], t$95$0, If[LessEqual[p$95$m, 1.65e-109], 1.0, If[LessEqual[p$95$m, 2.65e-59], t$95$0, If[LessEqual[p$95$m, 2e-42], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.06999999999999997e-159 or 1.64999999999999995e-109 < p < 2.6500000000000002e-59

   1. Initial program 72.3%

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

   3. Taylor expanded in x around -inf 17.4%

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

   4. Taylor expanded in p around -inf 21.0%

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

      1. neg-mul-1 21.0%

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

      2. distribute-neg-frac 21.0%

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

   6. Simplified 21.0%

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

   if 1.06999999999999997e-159 < p < 1.64999999999999995e-109 or 2.6500000000000002e-59 < p < 2.00000000000000008e-42

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-log-exp 74.6%

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

      2. +-commutative 74.6%

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

      3. add-sqr-sqrt 74.6%

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

      4. hypot-define 74.6%

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

      5. associate-*l* 74.6%

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

      6. sqrt-prod 74.6%

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

      7. metadata-eval 74.6%

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

      8. sqrt-unprod 74.6%

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

      9. add-sqr-sqrt 74.6%

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

   4. Applied egg-rr 74.6%

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

      1. add0 74.6%

         \[\leadsto \color{blue}{\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}\right)} + 0} \]

      2. rem-log-exp 74.6%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}} + 0 \]

      3. distribute-lft-in 74.6%

         \[\leadsto \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}}} + 0 \]

      4. metadata-eval 74.6%

         \[\leadsto \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0 \]

   6. Applied egg-rr 74.6%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0} \]
   7. Step-by-step derivation

      1. add0 74.6%

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

      2. *-commutative 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in x around inf 74.7%

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

   if 2.00000000000000008e-42 < p

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0 75.1%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.07 \cdot 10^{-159}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.65 \cdot 10^{-59}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 4.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 4.4e-13) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.4e-13) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 4.4d-13) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.4e-13) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 4.4e-13:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 4.4e-13)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 4.4e-13)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 4.4e-13], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 4.39999999999999993e-13

   1. Initial program 71.7%

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

   3. Taylor expanded in x around -inf 18.9%

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

   4. Taylor expanded in p around -inf 22.6%

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

      1. neg-mul-1 22.6%

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

      2. distribute-neg-frac 22.6%

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

   6. Simplified 22.6%

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

   if 4.39999999999999993e-13 < p

   1. Initial program 84.2%

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

   3. Taylor expanded in x around 0 78.8%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 29.0% accurate, 21.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1e-310) (/ p_m (- x)) (/ 1.0 (/ x p_m))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 / (x / p_m);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0 / (x / p_m)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 / (x / p_m);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1e-310:
		tmp = p_m / -x
	else:
		tmp = 1.0 / (x / p_m)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(1.0 / Float64(x / p_m));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = p_m / -x;
	else
		tmp = 1.0 / (x / p_m);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1e-310], N[(p$95$m / (-x)), $MachinePrecision], N[(1.0 / N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 51.5%

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

   3. Taylor expanded in x around -inf 32.8%

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

   4. Taylor expanded in p around -inf 38.6%

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

      1. neg-mul-1 38.6%

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

      2. distribute-neg-frac 38.6%

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

   6. Simplified 38.6%

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

   if -9.999999999999969e-311 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 4.5%

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

      1. associate-*r* 4.5%

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

      2. metadata-eval 4.5%

         \[\leadsto \sqrt{\color{blue}{1} \cdot \frac{{p}^{2}}{{x}^{2}}} \]

      3. *-un-lft-identity 4.5%

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

      4. div-inv 4.5%

         \[\leadsto \sqrt{\color{blue}{{p}^{2} \cdot \frac{1}{{x}^{2}}}} \]

      5. sqrt-prod 4.7%

         \[\leadsto \color{blue}{\sqrt{{p}^{2}} \cdot \sqrt{\frac{1}{{x}^{2}}}} \]

      6. unpow2 4.7%

         \[\leadsto \sqrt{\color{blue}{p \cdot p}} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      7. sqrt-prod 2.5%

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

      8. add-sqr-sqrt 3.4%

         \[\leadsto \color{blue}{p} \cdot \sqrt{\frac{1}{{x}^{2}}} \]

      9. metadata-eval 3.4%

         \[\leadsto p \cdot \sqrt{\frac{\color{blue}{1 \cdot 1}}{{x}^{2}}} \]

      10. unpow2 3.4%

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

      11. frac-times 3.4%

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

      12. metadata-eval 3.4%

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

      13. metadata-eval 3.4%

          \[\leadsto p \cdot \sqrt{\frac{\sqrt{\color{blue}{0.5 \cdot 2}}}{x} \cdot \frac{1}{x}} \]

      14. sqrt-unprod 3.4%

          \[\leadsto p \cdot \sqrt{\frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{2}}}{x} \cdot \frac{1}{x}} \]

      15. associate-*r/ 3.4%

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

      16. metadata-eval 3.4%

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

      17. metadata-eval 3.4%

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

      18. sqrt-unprod 3.4%

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

      19. associate-*r/ 3.4%

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

      20. sqr-neg 3.4%

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

   5. Applied egg-rr 3.5%

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

      1. un-div-inv 3.5%

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

      2. clear-num 3.5%

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

      3. add-sqr-sqrt 2.7%

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

      4. sqrt-unprod 4.7%

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

      5. sqr-neg 4.7%

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

      6. sqrt-unprod 2.5%

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

      7. add-sqr-sqrt 3.4%

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

   7. Applied egg-rr 3.4%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1e-310) (/ p_m (- x)) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d-310)) then
        tmp = p_m / -x
    else
        tmp = p_m / x
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1e-310) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1e-310:
		tmp = p_m / -x
	else:
		tmp = p_m / x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1e-310)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(p_m / x);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1e-310)
		tmp = p_m / -x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1e-310], N[(p$95$m / (-x)), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 51.5%

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

   3. Taylor expanded in x around -inf 32.8%

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

   4. Taylor expanded in p around -inf 38.6%

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

      1. neg-mul-1 38.6%

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

      2. distribute-neg-frac 38.6%

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

   6. Simplified 38.6%

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

   if -9.999999999999969e-311 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 4.5%

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

   4. Taylor expanded in p around 0 3.4%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 6.6% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Taylor expanded in x around -inf 19.3%

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

4. Taylor expanded in p around 0 17.2%

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

5. Final simplification 17.2%

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

Developer target: 79.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

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

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