Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.7%

Time: 11.3s

Alternatives: 7

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: 79.3% 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: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

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)))) <= -0.5) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(((1.0 + (x / hypot(x, (p_m * 2.0)))) * 0.5));
	}
	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)))) <= -0.5) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(((1.0 + (x / Math.hypot(x, (p_m * 2.0)))) * 0.5));
	}
	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)))) <= -0.5:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(((1.0 + (x / math.hypot(x, (p_m * 2.0)))) * 0.5))
	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)))) <= -0.5)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(Float64(1.0 + Float64(x / hypot(x, Float64(p_m * 2.0)))) * 0.5));
	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)))) <= -0.5)
		tmp = p_m / -x;
	else
		tmp = sqrt(((1.0 + (x / hypot(x, (p_m * 2.0)))) * 0.5));
	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], -0.5], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 15.1%

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

      1. +-commutative 15.1%

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

      2. sqr-neg 15.1%

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

      3. associate-*l* 15.1%

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

      4. sqr-neg 15.1%

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

      5. fma-define 15.1%

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

      6. sqr-neg 15.1%

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

      7. fma-define 15.1%

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

      8. associate-*l* 15.1%

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

      9. +-commutative 15.1%

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

   3. Simplified 15.4%

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

   5. Taylor expanded in x around -inf 60.7%

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

      1. mul-1-neg 60.7%

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

      2. distribute-neg-frac2 60.7%

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

   7. Simplified 60.7%

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

   if -0.5 < (/.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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. sqr-neg 100.0%

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

      7. fma-define 100.0%

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

      8. associate-*l* 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. fma-undefine 100.0%

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

      3. associate-*r* 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 90.9%

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

Alternative 2: 68.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ t_1 := 1 + \frac{-0.5 \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}\\ \mathbf{if}\;p\_m \leq 1.8 \cdot 10^{-258}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 3.2 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 5.2 \cdot 10^{-222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 3.1 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + 0.25 \cdot \frac{x}{p\_m}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))) (t_1 (+ 1.0 (/ (* -0.5 (* p_m p_m)) (* x x)))))
   (if (<= p_m 1.8e-258)
     t_1
     (if (<= p_m 3.2e-242)
       t_0
       (if (<= p_m 5.2e-222)
         t_1
         (if (<= p_m 1.55e-103)
           t_0
           (if (<= p_m 3.1e-32)
             1.0
             (cbrt (pow (+ 0.5 (* 0.25 (/ x p_m))) 1.5)))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 1.8e-258) {
		tmp = t_1;
	} else if (p_m <= 3.2e-242) {
		tmp = t_0;
	} else if (p_m <= 5.2e-222) {
		tmp = t_1;
	} else if (p_m <= 1.55e-103) {
		tmp = t_0;
	} else if (p_m <= 3.1e-32) {
		tmp = 1.0;
	} else {
		tmp = cbrt(pow((0.5 + (0.25 * (x / p_m))), 1.5));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 1.8e-258) {
		tmp = t_1;
	} else if (p_m <= 3.2e-242) {
		tmp = t_0;
	} else if (p_m <= 5.2e-222) {
		tmp = t_1;
	} else if (p_m <= 1.55e-103) {
		tmp = t_0;
	} else if (p_m <= 3.1e-32) {
		tmp = 1.0;
	} else {
		tmp = Math.cbrt(Math.pow((0.5 + (0.25 * (x / p_m))), 1.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	t_1 = Float64(1.0 + Float64(Float64(-0.5 * Float64(p_m * p_m)) / Float64(x * x)))
	tmp = 0.0
	if (p_m <= 1.8e-258)
		tmp = t_1;
	elseif (p_m <= 3.2e-242)
		tmp = t_0;
	elseif (p_m <= 5.2e-222)
		tmp = t_1;
	elseif (p_m <= 1.55e-103)
		tmp = t_0;
	elseif (p_m <= 3.1e-32)
		tmp = 1.0;
	else
		tmp = cbrt((Float64(0.5 + Float64(0.25 * Float64(x / p_m))) ^ 1.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(-0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 1.8e-258], t$95$1, If[LessEqual[p$95$m, 3.2e-242], t$95$0, If[LessEqual[p$95$m, 5.2e-222], t$95$1, If[LessEqual[p$95$m, 1.55e-103], t$95$0, If[LessEqual[p$95$m, 3.1e-32], 1.0, N[Power[N[Power[N[(0.5 + N[(0.25 * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 1.79999999999999989e-258 or 3.19999999999999999e-242 < p < 5.1999999999999997e-222

   1. Initial program 81.5%

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

      1. +-commutative 81.5%

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

      2. sqr-neg 81.5%

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

      3. associate-*l* 81.5%

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

      4. sqr-neg 81.5%

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

      5. fma-define 81.5%

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

      6. sqr-neg 81.5%

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

      7. fma-define 81.5%

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

      8. associate-*l* 81.5%

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

      9. +-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x around inf 37.0%

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

      1. associate-*r/ 37.0%

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

   7. Simplified 37.0%

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

      1. unpow2 37.0%

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

   9. Applied egg-rr 37.0%

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

       1. unpow2 37.0%

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

   11. Applied egg-rr 37.0%

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

   if 1.79999999999999989e-258 < p < 3.19999999999999999e-242 or 5.1999999999999997e-222 < p < 1.5500000000000001e-103

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

      4. sqr-neg 45.4%

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

      5. fma-define 45.4%

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

      6. sqr-neg 45.4%

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

      7. fma-define 45.4%

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

      8. associate-*l* 45.4%

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

      9. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in x around -inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. distribute-neg-frac2 66.6%

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

   7. Simplified 66.6%

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

   if 1.5500000000000001e-103 < p < 3.10000000000000011e-32

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 50.9%

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

   if 3.10000000000000011e-32 < p

   1. Initial program 93.1%

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

      1. +-commutative 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-*l* 93.1%

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

      4. sqr-neg 93.1%

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

      5. fma-define 93.1%

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

      6. sqr-neg 93.1%

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

      7. fma-define 93.1%

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

      8. associate-*l* 93.1%

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

      9. +-commutative 93.1%

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

   3. Simplified 93.3%

      \[\leadsto \color{blue}{\sqrt{0.5 + 0.5 \cdot \frac{x}{\sqrt{\mathsf{fma}\left(x, x, 4 \cdot \left(p \cdot p\right)\right)}}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 93.3%

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

      2. fma-undefine 93.1%

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

      3. associate-*r* 93.1%

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

      4. +-commutative 93.1%

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

      5. distribute-rgt1-in 93.1%

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

      6. +-commutative 93.1%

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

   6. Applied egg-rr 93.3%

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

      1. add-cbrt-cube 93.3%

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

      2. add-sqr-sqrt 93.3%

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

      3. pow1 93.3%

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

      4. pow1/2 93.3%

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

      5. pow-prod-up 93.3%

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

      6. *-commutative 93.3%

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

      7. distribute-lft-in 93.3%

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

      8. metadata-eval 93.3%

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

      9. metadata-eval 93.3%

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

   8. Applied egg-rr 93.3%

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

   9. Taylor expanded in x around 0 85.8%

      \[\leadsto \sqrt[3]{{\left(0.5 + \color{blue}{0.25 \cdot \frac{x}{p}}\right)}^{1.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.8 \cdot 10^{-258}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p \cdot p\right)}{x \cdot x}\\ \mathbf{elif}\;p \leq 3.2 \cdot 10^{-242}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 5.2 \cdot 10^{-222}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p \cdot p\right)}{x \cdot x}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.1 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(0.5 + 0.25 \cdot \frac{x}{p}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ t_1 := 1 + \frac{-0.5 \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}\\ \mathbf{if}\;p\_m \leq 9 \cdot 10^{-258}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 1.15 \cdot 10^{-241}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 6 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))) (t_1 (+ 1.0 (/ (* -0.5 (* p_m p_m)) (* x x)))))
   (if (<= p_m 9e-258)
     t_1
     (if (<= p_m 1.15e-241)
       t_0
       (if (<= p_m 1.55e-217)
         t_1
         (if (<= p_m 1.7e-103)
           t_0
           (if (<= p_m 6e-34) 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 t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 9e-258) {
		tmp = t_1;
	} else if (p_m <= 1.15e-241) {
		tmp = t_0;
	} else if (p_m <= 1.55e-217) {
		tmp = t_1;
	} else if (p_m <= 1.7e-103) {
		tmp = t_0;
	} else if (p_m <= 6e-34) {
		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) :: t_1
    real(8) :: tmp
    t_0 = p_m / -x
    t_1 = 1.0d0 + (((-0.5d0) * (p_m * p_m)) / (x * x))
    if (p_m <= 9d-258) then
        tmp = t_1
    else if (p_m <= 1.15d-241) then
        tmp = t_0
    else if (p_m <= 1.55d-217) then
        tmp = t_1
    else if (p_m <= 1.7d-103) then
        tmp = t_0
    else if (p_m <= 6d-34) 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 t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 9e-258) {
		tmp = t_1;
	} else if (p_m <= 1.15e-241) {
		tmp = t_0;
	} else if (p_m <= 1.55e-217) {
		tmp = t_1;
	} else if (p_m <= 1.7e-103) {
		tmp = t_0;
	} else if (p_m <= 6e-34) {
		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
	t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x))
	tmp = 0
	if p_m <= 9e-258:
		tmp = t_1
	elif p_m <= 1.15e-241:
		tmp = t_0
	elif p_m <= 1.55e-217:
		tmp = t_1
	elif p_m <= 1.7e-103:
		tmp = t_0
	elif p_m <= 6e-34:
		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))
	t_1 = Float64(1.0 + Float64(Float64(-0.5 * Float64(p_m * p_m)) / Float64(x * x)))
	tmp = 0.0
	if (p_m <= 9e-258)
		tmp = t_1;
	elseif (p_m <= 1.15e-241)
		tmp = t_0;
	elseif (p_m <= 1.55e-217)
		tmp = t_1;
	elseif (p_m <= 1.7e-103)
		tmp = t_0;
	elseif (p_m <= 6e-34)
		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;
	t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	tmp = 0.0;
	if (p_m <= 9e-258)
		tmp = t_1;
	elseif (p_m <= 1.15e-241)
		tmp = t_0;
	elseif (p_m <= 1.55e-217)
		tmp = t_1;
	elseif (p_m <= 1.7e-103)
		tmp = t_0;
	elseif (p_m <= 6e-34)
		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]}, Block[{t$95$1 = N[(1.0 + N[(N[(-0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 9e-258], t$95$1, If[LessEqual[p$95$m, 1.15e-241], t$95$0, If[LessEqual[p$95$m, 1.55e-217], t$95$1, If[LessEqual[p$95$m, 1.7e-103], t$95$0, If[LessEqual[p$95$m, 6e-34], 1.0, N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 9.00000000000000017e-258 or 1.15e-241 < p < 1.5499999999999999e-217

   1. Initial program 81.5%

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

      1. +-commutative 81.5%

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

      2. sqr-neg 81.5%

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

      3. associate-*l* 81.5%

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

      4. sqr-neg 81.5%

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

      5. fma-define 81.5%

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

      6. sqr-neg 81.5%

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

      7. fma-define 81.5%

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

      8. associate-*l* 81.5%

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

      9. +-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x around inf 37.0%

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

      1. associate-*r/ 37.0%

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

   7. Simplified 37.0%

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

      1. unpow2 37.0%

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

   9. Applied egg-rr 37.0%

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

       1. unpow2 37.0%

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

   11. Applied egg-rr 37.0%

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

   if 9.00000000000000017e-258 < p < 1.15e-241 or 1.5499999999999999e-217 < p < 1.70000000000000001e-103

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

      4. sqr-neg 45.4%

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

      5. fma-define 45.4%

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

      6. sqr-neg 45.4%

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

      7. fma-define 45.4%

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

      8. associate-*l* 45.4%

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

      9. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in x around -inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. distribute-neg-frac2 66.6%

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

   7. Simplified 66.6%

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

   if 1.70000000000000001e-103 < p < 6e-34

   1. Initial program 66.2%

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

   3. Taylor expanded in x around inf 50.9%

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

   if 6e-34 < p

   1. Initial program 93.1%

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

      1. +-commutative 93.1%

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

      2. sqr-neg 93.1%

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

      3. associate-*l* 93.1%

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

      4. sqr-neg 93.1%

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

      5. fma-define 93.1%

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

      6. sqr-neg 93.1%

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

      7. fma-define 93.1%

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

      8. associate-*l* 93.1%

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

      9. +-commutative 93.1%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in x around 0 85.8%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 9 \cdot 10^{-258}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p \cdot p\right)}{x \cdot x}\\ \mathbf{elif}\;p \leq 1.15 \cdot 10^{-241}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-217}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p \cdot p\right)}{x \cdot x}\\ \mathbf{elif}\;p \leq 1.7 \cdot 10^{-103}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 6 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ t_1 := 1 + \frac{-0.5 \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}\\ \mathbf{if}\;p\_m \leq 2.2 \cdot 10^{-258}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 3.5 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 9.5 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 8 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))) (t_1 (+ 1.0 (/ (* -0.5 (* p_m p_m)) (* x x)))))
   (if (<= p_m 2.2e-258)
     t_1
     (if (<= p_m 3.5e-242)
       t_0
       (if (<= p_m 1.55e-217)
         t_1
         (if (<= p_m 9.5e-104) t_0 (if (<= p_m 8e-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 t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 2.2e-258) {
		tmp = t_1;
	} else if (p_m <= 3.5e-242) {
		tmp = t_0;
	} else if (p_m <= 1.55e-217) {
		tmp = t_1;
	} else if (p_m <= 9.5e-104) {
		tmp = t_0;
	} else if (p_m <= 8e-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) :: t_1
    real(8) :: tmp
    t_0 = p_m / -x
    t_1 = 1.0d0 + (((-0.5d0) * (p_m * p_m)) / (x * x))
    if (p_m <= 2.2d-258) then
        tmp = t_1
    else if (p_m <= 3.5d-242) then
        tmp = t_0
    else if (p_m <= 1.55d-217) then
        tmp = t_1
    else if (p_m <= 9.5d-104) then
        tmp = t_0
    else if (p_m <= 8d-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 t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 2.2e-258) {
		tmp = t_1;
	} else if (p_m <= 3.5e-242) {
		tmp = t_0;
	} else if (p_m <= 1.55e-217) {
		tmp = t_1;
	} else if (p_m <= 9.5e-104) {
		tmp = t_0;
	} else if (p_m <= 8e-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
	t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x))
	tmp = 0
	if p_m <= 2.2e-258:
		tmp = t_1
	elif p_m <= 3.5e-242:
		tmp = t_0
	elif p_m <= 1.55e-217:
		tmp = t_1
	elif p_m <= 9.5e-104:
		tmp = t_0
	elif p_m <= 8e-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))
	t_1 = Float64(1.0 + Float64(Float64(-0.5 * Float64(p_m * p_m)) / Float64(x * x)))
	tmp = 0.0
	if (p_m <= 2.2e-258)
		tmp = t_1;
	elseif (p_m <= 3.5e-242)
		tmp = t_0;
	elseif (p_m <= 1.55e-217)
		tmp = t_1;
	elseif (p_m <= 9.5e-104)
		tmp = t_0;
	elseif (p_m <= 8e-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;
	t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	tmp = 0.0;
	if (p_m <= 2.2e-258)
		tmp = t_1;
	elseif (p_m <= 3.5e-242)
		tmp = t_0;
	elseif (p_m <= 1.55e-217)
		tmp = t_1;
	elseif (p_m <= 9.5e-104)
		tmp = t_0;
	elseif (p_m <= 8e-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]}, Block[{t$95$1 = N[(1.0 + N[(N[(-0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 2.2e-258], t$95$1, If[LessEqual[p$95$m, 3.5e-242], t$95$0, If[LessEqual[p$95$m, 1.55e-217], t$95$1, If[LessEqual[p$95$m, 9.5e-104], t$95$0, If[LessEqual[p$95$m, 8e-42], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 2.20000000000000015e-258 or 3.4999999999999999e-242 < p < 1.5499999999999999e-217

   1. Initial program 81.5%

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

      1. +-commutative 81.5%

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

      2. sqr-neg 81.5%

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

      3. associate-*l* 81.5%

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

      4. sqr-neg 81.5%

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

      5. fma-define 81.5%

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

      6. sqr-neg 81.5%

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

      7. fma-define 81.5%

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

      8. associate-*l* 81.5%

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

      9. +-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in x around inf 37.0%

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

      1. associate-*r/ 37.0%

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

   7. Simplified 37.0%

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

      1. unpow2 37.0%

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

   9. Applied egg-rr 37.0%

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

       1. unpow2 37.0%

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

   11. Applied egg-rr 37.0%

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

   if 2.20000000000000015e-258 < p < 3.4999999999999999e-242 or 1.5499999999999999e-217 < p < 9.5000000000000002e-104

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

      4. sqr-neg 45.4%

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

      5. fma-define 45.4%

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

      6. sqr-neg 45.4%

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

      7. fma-define 45.4%

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

      8. associate-*l* 45.4%

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

      9. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in x around -inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. distribute-neg-frac2 66.6%

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

   7. Simplified 66.6%

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

   if 9.5000000000000002e-104 < p < 8.0000000000000003e-42

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf 57.1%

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

   if 8.0000000000000003e-42 < p

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 82.9%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.2 \cdot 10^{-258}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p \cdot p\right)}{x \cdot x}\\ \mathbf{elif}\;p \leq 3.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-217}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p \cdot p\right)}{x \cdot x}\\ \mathbf{elif}\;p \leq 9.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 8 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ t_1 := 1 + \frac{-0.5 \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}\\ \mathbf{if}\;p\_m \leq 9 \cdot 10^{-259}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 9.6 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-218}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;p\_m \leq 7.8 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 3.2 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))) (t_1 (+ 1.0 (/ (* -0.5 (* p_m p_m)) (* x x)))))
   (if (<= p_m 9e-259)
     t_1
     (if (<= p_m 9.6e-242)
       t_0
       (if (<= p_m 1.55e-218)
         t_1
         (if (<= p_m 7.8e-104) t_0 (if (<= p_m 3.2e-42) t_1 (sqrt 0.5))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 9e-259) {
		tmp = t_1;
	} else if (p_m <= 9.6e-242) {
		tmp = t_0;
	} else if (p_m <= 1.55e-218) {
		tmp = t_1;
	} else if (p_m <= 7.8e-104) {
		tmp = t_0;
	} else if (p_m <= 3.2e-42) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = p_m / -x
    t_1 = 1.0d0 + (((-0.5d0) * (p_m * p_m)) / (x * x))
    if (p_m <= 9d-259) then
        tmp = t_1
    else if (p_m <= 9.6d-242) then
        tmp = t_0
    else if (p_m <= 1.55d-218) then
        tmp = t_1
    else if (p_m <= 7.8d-104) then
        tmp = t_0
    else if (p_m <= 3.2d-42) then
        tmp = t_1
    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 t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	double tmp;
	if (p_m <= 9e-259) {
		tmp = t_1;
	} else if (p_m <= 9.6e-242) {
		tmp = t_0;
	} else if (p_m <= 1.55e-218) {
		tmp = t_1;
	} else if (p_m <= 7.8e-104) {
		tmp = t_0;
	} else if (p_m <= 3.2e-42) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x))
	tmp = 0
	if p_m <= 9e-259:
		tmp = t_1
	elif p_m <= 9.6e-242:
		tmp = t_0
	elif p_m <= 1.55e-218:
		tmp = t_1
	elif p_m <= 7.8e-104:
		tmp = t_0
	elif p_m <= 3.2e-42:
		tmp = t_1
	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))
	t_1 = Float64(1.0 + Float64(Float64(-0.5 * Float64(p_m * p_m)) / Float64(x * x)))
	tmp = 0.0
	if (p_m <= 9e-259)
		tmp = t_1;
	elseif (p_m <= 9.6e-242)
		tmp = t_0;
	elseif (p_m <= 1.55e-218)
		tmp = t_1;
	elseif (p_m <= 7.8e-104)
		tmp = t_0;
	elseif (p_m <= 3.2e-42)
		tmp = t_1;
	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;
	t_1 = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	tmp = 0.0;
	if (p_m <= 9e-259)
		tmp = t_1;
	elseif (p_m <= 9.6e-242)
		tmp = t_0;
	elseif (p_m <= 1.55e-218)
		tmp = t_1;
	elseif (p_m <= 7.8e-104)
		tmp = t_0;
	elseif (p_m <= 3.2e-42)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(1.0 + N[(N[(-0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 9e-259], t$95$1, If[LessEqual[p$95$m, 9.6e-242], t$95$0, If[LessEqual[p$95$m, 1.55e-218], t$95$1, If[LessEqual[p$95$m, 7.8e-104], t$95$0, If[LessEqual[p$95$m, 3.2e-42], t$95$1, N[Sqrt[0.5], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 8.99999999999999947e-259 or 9.6000000000000004e-242 < p < 1.54999999999999999e-218 or 7.8000000000000004e-104 < p < 3.20000000000000025e-42

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. sqr-neg 80.8%

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

      3. associate-*l* 80.8%

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

      4. sqr-neg 80.8%

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

      5. fma-define 80.8%

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

      6. sqr-neg 80.8%

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

      7. fma-define 80.8%

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

      8. associate-*l* 80.8%

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

      9. +-commutative 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. associate-*r/ 38.8%

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

   7. Simplified 38.8%

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

      1. unpow2 38.8%

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

   9. Applied egg-rr 38.8%

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

       1. unpow2 38.8%

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

   11. Applied egg-rr 38.8%

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

   if 8.99999999999999947e-259 < p < 9.6000000000000004e-242 or 1.54999999999999999e-218 < p < 7.8000000000000004e-104

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

      2. sqr-neg 45.4%

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

      3. associate-*l* 45.4%

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

      4. sqr-neg 45.4%

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

      5. fma-define 45.4%

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

      6. sqr-neg 45.4%

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

      7. fma-define 45.4%

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

      8. associate-*l* 45.4%

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

      9. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in x around -inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. distribute-neg-frac2 66.6%

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

   7. Simplified 66.6%

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

   if 3.20000000000000025e-42 < p

   1. Initial program 90.9%

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

   3. Taylor expanded in x around 0 82.9%

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

Alternative 6: 50.9% accurate, 13.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-150}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.5 \cdot \left(p\_m \cdot p\_m\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x 1.22e-150) (/ p_m (- x)) (+ 1.0 (/ (* -0.5 (* p_m p_m)) (* x x)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= 1.22e-150) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 + ((-0.5 * (p_m * p_m)) / (x * 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 <= 1.22d-150) then
        tmp = p_m / -x
    else
        tmp = 1.0d0 + (((-0.5d0) * (p_m * p_m)) / (x * 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 <= 1.22e-150) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= 1.22e-150:
		tmp = p_m / -x
	else:
		tmp = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= 1.22e-150)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(1.0 + Float64(Float64(-0.5 * Float64(p_m * p_m)) / Float64(x * x)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= 1.22e-150)
		tmp = p_m / -x;
	else
		tmp = 1.0 + ((-0.5 * (p_m * p_m)) / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, 1.22e-150], N[(p$95$m / (-x)), $MachinePrecision], N[(1.0 + N[(N[(-0.5 * N[(p$95$m * p$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.22e-150

   1. Initial program 59.0%

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

      1. +-commutative 59.0%

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

      2. sqr-neg 59.0%

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

      3. associate-*l* 59.0%

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

      4. sqr-neg 59.0%

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

      5. fma-define 59.0%

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

      6. sqr-neg 59.0%

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

      7. fma-define 59.0%

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

      8. associate-*l* 59.0%

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

      9. +-commutative 59.0%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in x around -inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. distribute-neg-frac2 31.3%

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

   7. Simplified 31.3%

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

   if 1.22e-150 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-define 100.0%

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

      6. sqr-neg 100.0%

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

      7. fma-define 100.0%

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

      8. associate-*l* 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 51.5%

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

      1. associate-*r/ 51.5%

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

   7. Simplified 51.5%

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

      1. unpow2 51.5%

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

   9. Applied egg-rr 51.5%

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

       1. unpow2 51.5%

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

   11. Applied egg-rr 51.5%

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

Alternative 7: 26.8% accurate, 53.8× 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 / Float64(-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 80.4%

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

   1. +-commutative 80.4%

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

   2. sqr-neg 80.4%

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

   3. associate-*l* 80.4%

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

   4. sqr-neg 80.4%

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

   5. fma-define 80.4%

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

   6. sqr-neg 80.4%

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

   7. fma-define 80.4%

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

   8. associate-*l* 80.4%

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

   9. +-commutative 80.4%

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

3. Simplified 80.5%

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

5. Taylor expanded in x around -inf 16.7%

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

   1. mul-1-neg 16.7%

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

   2. distribute-neg-frac2 16.7%

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

7. Simplified 16.7%

   \[\leadsto \color{blue}{\frac{p}{-x}} \]
8. Add Preprocessing

Developer target: 79.3% 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 2024085 
(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))))))))