Given's Rotation SVD example

Percentage Accurate: 78.7% → 99.8%

Time: 10.7s

Alternatives: 6

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: 78.7% 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

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ p_m (- x))
   (sqrt (* 0.5 (exp (log1p (/ 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)))) <= -1.0) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 * exp(log1p((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)))) <= -1.0) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 * Math.exp(Math.log1p((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)))) <= -1.0:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 * math.exp(math.log1p((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)))) <= -1.0)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * exp(log1p(Float64(x / hypot(x, Float64(p_m * 2.0)))))));
	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], -1.0], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[Exp[N[Log[1 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 21.4%

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

      1. add-exp-log 21.4%

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

      2. log1p-define 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 20.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 21.4%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 10.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 21.4%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 21.4%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 21.4%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 21.4%

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

      3. rem-exp-log 21.4%

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

      4. hypot-undefine 21.4%

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

      5. +-commutative 21.4%

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

      6. hypot-undefine 21.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 21.4%

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

      9. sqrt-prod 21.4%

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

      10. add-sqr-sqrt 21.4%

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

      11. pow2 21.4%

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

   6. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around -inf 62.1%

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

      1. neg-mul-1 62.1%

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

      2. distribute-neg-frac 62.1%

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

   9. Simplified 62.1%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. log1p-define 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 100.0%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 49.7%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -1

   1. Initial program 21.4%

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

      1. add-exp-log 21.4%

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

      2. log1p-define 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 20.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 21.4%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 21.4%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 10.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 21.4%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 21.4%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 21.4%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 21.4%

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

      3. rem-exp-log 21.4%

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

      4. hypot-undefine 21.4%

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

      5. +-commutative 21.4%

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

      6. hypot-undefine 21.4%

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

      7. add-sqr-sqrt 0.0%

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

      8. add-sqr-sqrt 21.4%

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

      9. sqrt-prod 21.4%

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

      10. add-sqr-sqrt 21.4%

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

      11. pow2 21.4%

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

   6. Applied egg-rr 21.4%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around -inf 62.1%

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

      1. neg-mul-1 62.1%

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

      2. distribute-neg-frac 62.1%

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

   9. Simplified 62.1%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 49.7%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 89.5%

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

Alternative 3: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 2.25 \cdot 10^{-296}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.22 \cdot 10^{-215}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.6 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 2.25e-296)
     1.0
     (if (<= p_m 1.22e-215)
       t_0
       (if (<= p_m 1.6e-131)
         1.0
         (if (<= p_m 2e-100) t_0 (if (<= p_m 1.55e-35) 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 <= 2.25e-296) {
		tmp = 1.0;
	} else if (p_m <= 1.22e-215) {
		tmp = t_0;
	} else if (p_m <= 1.6e-131) {
		tmp = 1.0;
	} else if (p_m <= 2e-100) {
		tmp = t_0;
	} else if (p_m <= 1.55e-35) {
		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 <= 2.25d-296) then
        tmp = 1.0d0
    else if (p_m <= 1.22d-215) then
        tmp = t_0
    else if (p_m <= 1.6d-131) then
        tmp = 1.0d0
    else if (p_m <= 2d-100) then
        tmp = t_0
    else if (p_m <= 1.55d-35) 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 <= 2.25e-296) {
		tmp = 1.0;
	} else if (p_m <= 1.22e-215) {
		tmp = t_0;
	} else if (p_m <= 1.6e-131) {
		tmp = 1.0;
	} else if (p_m <= 2e-100) {
		tmp = t_0;
	} else if (p_m <= 1.55e-35) {
		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 <= 2.25e-296:
		tmp = 1.0
	elif p_m <= 1.22e-215:
		tmp = t_0
	elif p_m <= 1.6e-131:
		tmp = 1.0
	elif p_m <= 2e-100:
		tmp = t_0
	elif p_m <= 1.55e-35:
		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 <= 2.25e-296)
		tmp = 1.0;
	elseif (p_m <= 1.22e-215)
		tmp = t_0;
	elseif (p_m <= 1.6e-131)
		tmp = 1.0;
	elseif (p_m <= 2e-100)
		tmp = t_0;
	elseif (p_m <= 1.55e-35)
		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 <= 2.25e-296)
		tmp = 1.0;
	elseif (p_m <= 1.22e-215)
		tmp = t_0;
	elseif (p_m <= 1.6e-131)
		tmp = 1.0;
	elseif (p_m <= 2e-100)
		tmp = t_0;
	elseif (p_m <= 1.55e-35)
		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, 2.25e-296], 1.0, If[LessEqual[p$95$m, 1.22e-215], t$95$0, If[LessEqual[p$95$m, 1.6e-131], 1.0, If[LessEqual[p$95$m, 2e-100], t$95$0, If[LessEqual[p$95$m, 1.55e-35], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 2.2500000000000001e-296 or 1.2199999999999999e-215 < p < 1.6e-131 or 2e-100 < p < 1.55000000000000006e-35

   1. Initial program 77.6%

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

      1. add-exp-log 77.6%

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

      2. log1p-define 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 77.7%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 77.6%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 77.6%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 13.1%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 77.6%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 77.6%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 77.0%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 77.0%

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

      3. rem-exp-log 77.0%

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

      4. hypot-undefine 77.0%

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

      5. +-commutative 77.0%

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

      6. hypot-undefine 77.0%

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

      7. add-sqr-sqrt 55.8%

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

      8. add-sqr-sqrt 77.0%

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

      9. sqrt-prod 77.6%

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

      10. add-sqr-sqrt 77.0%

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

      11. pow2 77.0%

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

   6. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around inf 44.3%

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

   if 2.2500000000000001e-296 < p < 1.2199999999999999e-215 or 1.6e-131 < p < 2e-100

   1. Initial program 46.8%

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

      1. add-exp-log 46.8%

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

      2. log1p-define 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 44.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 46.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 46.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 46.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 46.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 46.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 46.5%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 46.5%

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

      3. rem-exp-log 46.5%

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

      4. hypot-undefine 46.5%

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

      5. +-commutative 46.5%

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

      6. hypot-undefine 46.5%

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

      7. add-sqr-sqrt 24.6%

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

      8. add-sqr-sqrt 46.5%

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

      9. sqrt-prod 46.8%

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

      10. add-sqr-sqrt 46.8%

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

      11. pow2 46.8%

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

   6. Applied egg-rr 46.8%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around -inf 75.7%

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

      1. neg-mul-1 75.7%

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

      2. distribute-neg-frac 75.7%

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

   9. Simplified 75.7%

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

   if 1.55000000000000006e-35 < p

   1. Initial program 94.3%

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

   3. Taylor expanded in x around 0 92.2%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.25 \cdot 10^{-296}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.22 \cdot 10^{-215}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-131}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2 \cdot 10^{-100}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.9% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-149}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.3e-149) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.3e-149) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d-149)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1.3e-149) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.3e-149:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.3e-149)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.3e-149)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.3e-149], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.29999999999999999e-149

   1. Initial program 59.2%

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

      1. add-exp-log 59.2%

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

      2. log1p-define 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 58.7%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 59.2%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 59.2%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 34.0%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 59.2%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 59.2%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 59.2%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 59.2%

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

      3. rem-exp-log 59.2%

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

      4. hypot-undefine 59.2%

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

      5. +-commutative 59.2%

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

      6. hypot-undefine 59.2%

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

      7. add-sqr-sqrt 4.4%

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

      8. add-sqr-sqrt 59.2%

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

      9. sqrt-prod 59.2%

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

      10. add-sqr-sqrt 58.5%

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

      11. pow2 58.5%

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

   6. Applied egg-rr 58.5%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around -inf 34.0%

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

      1. neg-mul-1 34.0%

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

      2. distribute-neg-frac 34.0%

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

   9. Simplified 34.0%

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

   if -1.29999999999999999e-149 < x

   1. Initial program 100.0%

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

      1. add-exp-log 100.0%

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

      2. log1p-define 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 100.0%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 100.0%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 44.5%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 100.0%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 99.1%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 99.1%

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

      3. rem-exp-log 99.1%

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

      4. hypot-undefine 99.1%

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

      5. +-commutative 99.1%

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

      6. hypot-undefine 99.1%

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

      7. add-sqr-sqrt 99.1%

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

      8. add-sqr-sqrt 99.1%

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

      9. sqrt-prod 100.0%

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

      10. add-sqr-sqrt 99.3%

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

      11. pow2 99.3%

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

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around inf 62.9%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-149}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 36.9% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.5e+69) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.5e+69) {
		tmp = p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.5d+69)) then
        tmp = p_m / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1.5e+69) {
		tmp = p_m / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.5e+69:
		tmp = p_m / x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.5e+69)
		tmp = Float64(p_m / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.5e+69)
		tmp = p_m / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.5e+69], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999992e69

   1. Initial program 56.1%

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

      1. add-exp-log 56.1%

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

      2. log1p-define 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 54.3%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 56.1%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 56.1%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 32.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 56.1%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 56.1%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

   5. Taylor expanded in x around -inf 49.0%

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

      1. mul-1-neg 49.0%

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

      2. associate-/l* 49.0%

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

      3. distribute-rgt-neg-in 49.0%

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

      4. *-commutative 49.0%

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

      5. associate-/l* 49.1%

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

   7. Simplified 49.1%

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

      1. add-sqr-sqrt 49.2%

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

      2. sqrt-unprod 49.1%

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

      3. sqr-neg 49.1%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 53.3%

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

      6. associate-*r/ 53.1%

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

      7. sqrt-unprod 53.5%

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

      8. metadata-eval 53.5%

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

      9. metadata-eval 53.5%

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

      10. associate-*r/ 53.7%

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

      11. *-commutative 53.7%

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

      12. *-un-lft-identity 53.7%

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

   9. Applied egg-rr 53.7%

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

   if -1.49999999999999992e69 < x

   1. Initial program 82.8%

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

      1. add-exp-log 82.8%

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

      2. log1p-define 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      3. div-inv 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      4. div-inv 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

      5. +-commutative 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

      6. add-sqr-sqrt 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

      7. hypot-define 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

      8. associate-*l* 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

      9. sqrt-prod 82.8%

         \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

      10. metadata-eval 82.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

      11. sqrt-unprod 40.2%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

      12. add-sqr-sqrt 82.8%

          \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

   4. Applied egg-rr 82.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
   5. Step-by-step derivation

      1. sqrt-prod 82.3%

         \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

      2. log1p-undefine 82.3%

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

      3. rem-exp-log 82.3%

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

      4. hypot-undefine 82.3%

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

      5. +-commutative 82.3%

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

      6. hypot-undefine 82.3%

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

      7. add-sqr-sqrt 58.5%

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

      8. add-sqr-sqrt 82.3%

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

      9. sqrt-prod 82.8%

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

      10. add-sqr-sqrt 82.0%

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

      11. pow2 82.0%

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

   6. Applied egg-rr 82.0%

      \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

   7. Taylor expanded in x around inf 41.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.6% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.2%

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

   1. add-exp-log 78.2%

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

   2. log1p-define 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\color{blue}{\mathsf{log1p}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   3. div-inv 77.9%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{x \cdot \frac{1}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

   4. div-inv 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]

   5. +-commutative 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\color{blue}{x \cdot x + \left(4 \cdot p\right) \cdot p}}}\right)}} \]

   6. add-sqr-sqrt 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\sqrt{x \cdot x + \color{blue}{\sqrt{\left(4 \cdot p\right) \cdot p} \cdot \sqrt{\left(4 \cdot p\right) \cdot p}}}}\right)}} \]

   7. hypot-define 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\mathsf{hypot}\left(x, \sqrt{\left(4 \cdot p\right) \cdot p}\right)}}\right)}} \]

   8. associate-*l* 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \sqrt{\color{blue}{4 \cdot \left(p \cdot p\right)}}\right)}\right)}} \]

   9. sqrt-prod 78.2%

      \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{\sqrt{4} \cdot \sqrt{p \cdot p}}\right)}\right)}} \]

   10. metadata-eval 78.2%

       \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)}} \]

   11. sqrt-unprod 38.9%

       \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)}} \]

   12. add-sqr-sqrt 78.2%

       \[\leadsto \sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)}} \]

4. Applied egg-rr 78.2%

   \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]
5. Step-by-step derivation

   1. sqrt-prod 77.8%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}}} \]

   2. log1p-undefine 77.8%

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

   3. rem-exp-log 77.8%

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

   4. hypot-undefine 77.8%

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

   5. +-commutative 77.8%

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

   6. hypot-undefine 77.8%

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

   7. add-sqr-sqrt 48.4%

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

   8. add-sqr-sqrt 77.8%

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

   9. sqrt-prod 78.2%

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

   10. add-sqr-sqrt 77.5%

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

   11. pow2 77.5%

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

6. Applied egg-rr 77.5%

   \[\leadsto \color{blue}{{\left({\left(0.5 + \frac{0.5 \cdot x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}^{0.25}\right)}^{2}} \]

7. Taylor expanded in x around inf 35.7%

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

8. Final simplification 35.7%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 78.7% 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 2024079 
(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))))))))