Given's Rotation SVD example

Percentage Accurate: 78.8% → 99.6%

Time: 13.3s

Alternatives: 7

Speedup: 1.0×

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: 78.8% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{1.5 \cdot \frac{{p\_m}^{3}}{{x}^{2}} - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.98)
   (/ (- (* 1.5 (/ (pow p_m 3.0) (pow x 2.0))) p_m) x)
   (sqrt (fma x (/ 0.5 (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)))) <= -0.98) {
		tmp = ((1.5 * (pow(p_m, 3.0) / pow(x, 2.0))) - p_m) / x;
	} else {
		tmp = sqrt(fma(x, (0.5 / 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)))) <= -0.98)
		tmp = Float64(Float64(Float64(1.5 * Float64((p_m ^ 3.0) / (x ^ 2.0))) - p_m) / x);
	else
		tmp = sqrt(fma(x, Float64(0.5 / hypot(x, Float64(p_m * 2.0))), 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(N[(1.5 * N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - p$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $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)))) < -0.97999999999999998

   1. Initial program 14.4%

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

      1. add-sqr-sqrt 14.4%

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

      2. pow2 14.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 14.4%

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

   5. Taylor expanded in x around -inf 49.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 49.1%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-neg-frac2 49.1%

         \[\leadsto \color{blue}{\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{-x}} \]

      3. associate-*r/ 49.1%

         \[\leadsto \frac{p + \color{blue}{\frac{0.125 \cdot \left(-16 \cdot {p}^{4} + 4 \cdot {p}^{4}\right)}{p \cdot {x}^{2}}}}{-x} \]

      4. distribute-rgt-out 49.2%

         \[\leadsto \frac{p + \frac{0.125 \cdot \color{blue}{\left({p}^{4} \cdot \left(-16 + 4\right)\right)}}{p \cdot {x}^{2}}}{-x} \]

      5. metadata-eval 49.2%

         \[\leadsto \frac{p + \frac{0.125 \cdot \left({p}^{4} \cdot \color{blue}{-12}\right)}{p \cdot {x}^{2}}}{-x} \]

      6. *-commutative 49.2%

         \[\leadsto \frac{p + \frac{0.125 \cdot \color{blue}{\left(-12 \cdot {p}^{4}\right)}}{p \cdot {x}^{2}}}{-x} \]

      7. associate-*r* 49.2%

         \[\leadsto \frac{p + \frac{\color{blue}{\left(0.125 \cdot -12\right) \cdot {p}^{4}}}{p \cdot {x}^{2}}}{-x} \]

      8. metadata-eval 49.2%

         \[\leadsto \frac{p + \frac{\color{blue}{-1.5} \cdot {p}^{4}}{p \cdot {x}^{2}}}{-x} \]

   7. Simplified 49.2%

      \[\leadsto \color{blue}{\frac{p + \frac{-1.5 \cdot {p}^{4}}{p \cdot {x}^{2}}}{-x}} \]

   8. Taylor expanded in x around inf 59.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot p + 1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. +-commutative 59.4%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} + -1 \cdot p}}{x} \]

      2. mul-1-neg 59.4%

         \[\leadsto \frac{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} + \color{blue}{\left(-p\right)}}{x} \]

      3. unsub-neg 59.4%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} - p}}{x} \]

   10. Simplified 59.4%

       \[\leadsto \color{blue}{\frac{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} - p}{x}} \]

   if -0.97999999999999998 < (/.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 98.9%

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

      2. pow2 98.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 98.9%

         \[\leadsto \color{blue}{1 \cdot {\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]

      2. pow-pow 100.0%

         \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{\left(0.25 \cdot 2\right)}} \]

      3. metadata-eval 100.0%

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

      4. pow1/2 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. fma-undefine 100.0%

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

      3. associate-*l/ 100.0%

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

      4. associate-/l* 100.0%

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

      5. fma-define 100.0%

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

      6. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{1.5 \cdot \frac{{p\_m}^{3}}{{x}^{2}} - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.98)
   (/ (- (* 1.5 (/ (pow p_m 3.0) (pow x 2.0))) p_m) x)
   (sqrt (* 0.5 (+ -1.0 (+ 2.0 (/ x (hypot x (* p_m 2.0)))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98) {
		tmp = ((1.5 * (pow(p_m, 3.0) / pow(x, 2.0))) - p_m) / x;
	} else {
		tmp = sqrt((0.5 * (-1.0 + (2.0 + (x / hypot(x, (p_m * 2.0)))))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98) {
		tmp = ((1.5 * (Math.pow(p_m, 3.0) / Math.pow(x, 2.0))) - p_m) / x;
	} else {
		tmp = Math.sqrt((0.5 * (-1.0 + (2.0 + (x / Math.hypot(x, (p_m * 2.0)))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98:
		tmp = ((1.5 * (math.pow(p_m, 3.0) / math.pow(x, 2.0))) - p_m) / x
	else:
		tmp = math.sqrt((0.5 * (-1.0 + (2.0 + (x / math.hypot(x, (p_m * 2.0)))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.98)
		tmp = Float64(Float64(Float64(1.5 * Float64((p_m ^ 3.0) / (x ^ 2.0))) - p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(-1.0 + Float64(2.0 + Float64(x / hypot(x, Float64(p_m * 2.0)))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.98)
		tmp = ((1.5 * ((p_m ^ 3.0) / (x ^ 2.0))) - p_m) / x;
	else
		tmp = sqrt((0.5 * (-1.0 + (2.0 + (x / hypot(x, (p_m * 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.98], N[(N[(N[(1.5 * N[(N[Power[p$95$m, 3.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - p$95$m), $MachinePrecision] / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(-1.0 + N[(2.0 + 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)))) < -0.97999999999999998

   1. Initial program 14.4%

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

      1. add-sqr-sqrt 14.4%

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

      2. pow2 14.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 14.4%

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

   5. Taylor expanded in x around -inf 49.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 49.1%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-neg-frac2 49.1%

         \[\leadsto \color{blue}{\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{-x}} \]

      3. associate-*r/ 49.1%

         \[\leadsto \frac{p + \color{blue}{\frac{0.125 \cdot \left(-16 \cdot {p}^{4} + 4 \cdot {p}^{4}\right)}{p \cdot {x}^{2}}}}{-x} \]

      4. distribute-rgt-out 49.2%

         \[\leadsto \frac{p + \frac{0.125 \cdot \color{blue}{\left({p}^{4} \cdot \left(-16 + 4\right)\right)}}{p \cdot {x}^{2}}}{-x} \]

      5. metadata-eval 49.2%

         \[\leadsto \frac{p + \frac{0.125 \cdot \left({p}^{4} \cdot \color{blue}{-12}\right)}{p \cdot {x}^{2}}}{-x} \]

      6. *-commutative 49.2%

         \[\leadsto \frac{p + \frac{0.125 \cdot \color{blue}{\left(-12 \cdot {p}^{4}\right)}}{p \cdot {x}^{2}}}{-x} \]

      7. associate-*r* 49.2%

         \[\leadsto \frac{p + \frac{\color{blue}{\left(0.125 \cdot -12\right) \cdot {p}^{4}}}{p \cdot {x}^{2}}}{-x} \]

      8. metadata-eval 49.2%

         \[\leadsto \frac{p + \frac{\color{blue}{-1.5} \cdot {p}^{4}}{p \cdot {x}^{2}}}{-x} \]

   7. Simplified 49.2%

      \[\leadsto \color{blue}{\frac{p + \frac{-1.5 \cdot {p}^{4}}{p \cdot {x}^{2}}}{-x}} \]

   8. Taylor expanded in x around inf 59.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot p + 1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. +-commutative 59.4%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} + -1 \cdot p}}{x} \]

      2. mul-1-neg 59.4%

         \[\leadsto \frac{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} + \color{blue}{\left(-p\right)}}{x} \]

      3. unsub-neg 59.4%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} - p}}{x} \]

   10. Simplified 59.4%

       \[\leadsto \color{blue}{\frac{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} - p}{x}} \]

   if -0.97999999999999998 < (/.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. expm1-log1p-u 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}} \]

      2. expm1-undefine 99.5%

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

      3. +-commutative 99.5%

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

      4. add-sqr-sqrt 99.5%

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

      5. hypot-define 99.5%

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

      6. associate-*l* 99.5%

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

      7. sqrt-prod 99.5%

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

      8. metadata-eval 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, \color{blue}{2} \cdot \sqrt{p \cdot p}\right)}\right)} - 1\right)} \]

      9. sqrt-unprod 44.6%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{\left(\sqrt{p} \cdot \sqrt{p}\right)}\right)}\right)} - 1\right)} \]

      10. add-sqr-sqrt 99.5%

          \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot \color{blue}{p}\right)}\right)} - 1\right)} \]

   4. Applied egg-rr 99.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} - 1\right)}} \]
   5. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \left(-1\right)\right)}} \]

      2. metadata-eval 99.5%

         \[\leadsto \sqrt{0.5 \cdot \left(e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)} + \color{blue}{-1}\right)} \]

      3. +-commutative 99.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(1 + \frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}\right)}\right)}} \]

      4. log1p-undefine 100.0%

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

      5. rem-exp-log 100.0%

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

      6. associate-+r+ 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;\frac{1.5 \cdot \frac{{p}^{3}}{{x}^{2}} - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1.25e+27)
   (/ 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 <= -1.25e+27) {
		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 <= -1.25e+27) {
		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 <= -1.25e+27:
		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 (x <= -1.25e+27)
		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 <= -1.25e+27)
		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[x, -1.25e+27], 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 x < -1.24999999999999995e27

   1. Initial program 44.8%

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

      1. add-sqr-sqrt 44.4%

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

      2. pow2 44.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 44.4%

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

   5. Taylor expanded in x around -inf 43.4%

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

      1. mul-1-neg 43.4%

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

      2. distribute-neg-frac2 43.4%

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

   7. Simplified 43.4%

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

   if -1.24999999999999995e27 < x

   1. Initial program 88.5%

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

      1. add-sqr-sqrt 88.5%

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

      2. hypot-define 88.5%

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

      3. associate-*l* 88.5%

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

      4. sqrt-prod 88.5%

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

      5. metadata-eval 88.5%

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

      6. sqrt-unprod 39.7%

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

      7. add-sqr-sqrt 88.5%

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

   4. Applied egg-rr 88.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := 1 + \left(\frac{p\_m}{x} \cdot \frac{p\_m}{x}\right) \cdot -0.5\\ \mathbf{if}\;p\_m \leq 6.2 \cdot 10^{-210}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))))
   (if (<= p_m 6.2e-210)
     t_0
     (if (<= p_m 1.6e-133)
       (/ p_m (- x))
       (if (<= p_m 6.5e-71) t_0 (sqrt 0.5))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	double tmp;
	if (p_m <= 6.2e-210) {
		tmp = t_0;
	} else if (p_m <= 1.6e-133) {
		tmp = p_m / -x;
	} else if (p_m <= 6.5e-71) {
		tmp = t_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 = 1.0d0 + (((p_m / x) * (p_m / x)) * (-0.5d0))
    if (p_m <= 6.2d-210) then
        tmp = t_0
    else if (p_m <= 1.6d-133) then
        tmp = p_m / -x
    else if (p_m <= 6.5d-71) then
        tmp = t_0
    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 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	double tmp;
	if (p_m <= 6.2e-210) {
		tmp = t_0;
	} else if (p_m <= 1.6e-133) {
		tmp = p_m / -x;
	} else if (p_m <= 6.5e-71) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	tmp = 0
	if p_m <= 6.2e-210:
		tmp = t_0
	elif p_m <= 1.6e-133:
		tmp = p_m / -x
	elif p_m <= 6.5e-71:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5))
	tmp = 0.0
	if (p_m <= 6.2e-210)
		tmp = t_0;
	elseif (p_m <= 1.6e-133)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 6.5e-71)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	tmp = 0.0;
	if (p_m <= 6.2e-210)
		tmp = t_0;
	elseif (p_m <= 1.6e-133)
		tmp = p_m / -x;
	elseif (p_m <= 6.5e-71)
		tmp = t_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[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 6.2e-210], t$95$0, If[LessEqual[p$95$m, 1.6e-133], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 6.5e-71], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 6.19999999999999973e-210 or 1.60000000000000006e-133 < p < 6.50000000000000005e-71

   1. Initial program 79.3%

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

      1. add-sqr-sqrt 78.6%

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

      2. pow2 78.6%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in x around inf 30.2%

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

      1. *-commutative 30.2%

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

   7. Simplified 30.2%

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

      1. unpow2 30.2%

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

      2. unpow2 30.2%

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

      3. times-frac 30.2%

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

   9. Applied egg-rr 30.2%

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

   if 6.19999999999999973e-210 < p < 1.60000000000000006e-133

   1. Initial program 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 49.7%

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

   5. Taylor expanded in x around -inf 58.7%

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

      1. mul-1-neg 58.7%

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

      2. distribute-neg-frac2 58.7%

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

   7. Simplified 58.7%

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

   if 6.50000000000000005e-71 < p

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0 79.2%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 6.2 \cdot 10^{-210}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 5 \cdot 10^{-210}:\\ \;\;\;\;1 + \left(\frac{p\_m}{x} \cdot \frac{p\_m}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 7.8 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 5e-210)
   (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))
   (if (<= p_m 1.55e-133) (/ p_m (- x)) (if (<= p_m 7.8e-71) 1.0 (sqrt 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 5e-210) {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	} else if (p_m <= 1.55e-133) {
		tmp = p_m / -x;
	} else if (p_m <= 7.8e-71) {
		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) :: tmp
    if (p_m <= 5d-210) then
        tmp = 1.0d0 + (((p_m / x) * (p_m / x)) * (-0.5d0))
    else if (p_m <= 1.55d-133) then
        tmp = p_m / -x
    else if (p_m <= 7.8d-71) 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 tmp;
	if (p_m <= 5e-210) {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	} else if (p_m <= 1.55e-133) {
		tmp = p_m / -x;
	} else if (p_m <= 7.8e-71) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 5e-210:
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	elif p_m <= 1.55e-133:
		tmp = p_m / -x
	elif p_m <= 7.8e-71:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 5e-210)
		tmp = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5));
	elseif (p_m <= 1.55e-133)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 7.8e-71)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 5e-210)
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	elseif (p_m <= 1.55e-133)
		tmp = p_m / -x;
	elseif (p_m <= 7.8e-71)
		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_] := If[LessEqual[p$95$m, 5e-210], N[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 1.55e-133], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 7.8e-71], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if p < 5.0000000000000002e-210

   1. Initial program 79.2%

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

      1. add-sqr-sqrt 78.5%

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

      2. pow2 78.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 78.5%

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

   5. Taylor expanded in x around inf 26.3%

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

      1. *-commutative 26.3%

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

   7. Simplified 26.3%

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

      1. unpow2 26.3%

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

      2. unpow2 26.3%

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

      3. times-frac 26.3%

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

   9. Applied egg-rr 26.3%

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

   if 5.0000000000000002e-210 < p < 1.55000000000000008e-133

   1. Initial program 49.7%

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

      1. add-sqr-sqrt 49.7%

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

      2. pow2 49.7%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 49.7%

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

   5. Taylor expanded in x around -inf 58.7%

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

      1. mul-1-neg 58.7%

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

      2. distribute-neg-frac2 58.7%

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

   7. Simplified 58.7%

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

   if 1.55000000000000008e-133 < p < 7.8000000000000004e-71

   1. Initial program 79.6%

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

      1. add-sqr-sqrt 79.5%

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

      2. pow2 79.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 79.5%

         \[\leadsto \color{blue}{1 \cdot {\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{0.25}\right)}^{2}} \]

      2. pow-pow 79.6%

         \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{\left(0.25 \cdot 2\right)}} \]

      3. metadata-eval 79.6%

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

      4. pow1/2 79.6%

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

   6. Applied egg-rr 79.6%

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

      1. *-lft-identity 79.6%

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

      2. fma-undefine 79.6%

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

      3. associate-*l/ 79.6%

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

      4. associate-/l* 79.7%

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

      5. fma-define 79.0%

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

      6. *-commutative 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in x around inf 73.9%

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

   if 7.8000000000000004e-71 < p

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0 79.2%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 5 \cdot 10^{-210}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-133}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 7.8 \cdot 10^{-71}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-150}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{p\_m}{x} \cdot \frac{p\_m}{x}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x 1.55e-150) (/ p_m (- x)) (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= 1.55e-150) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d-150) then
        tmp = p_m / -x
    else
        tmp = 1.0d0 + (((p_m / x) * (p_m / x)) * (-0.5d0))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= 1.55e-150) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= 1.55e-150:
		tmp = p_m / -x
	else:
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= 1.55e-150)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= 1.55e-150)
		tmp = p_m / -x;
	else
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, 1.55e-150], N[(p$95$m / (-x)), $MachinePrecision], N[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999999e-150

   1. Initial program 63.0%

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

      1. add-sqr-sqrt 62.2%

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

      2. pow2 62.2%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 62.2%

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

   5. Taylor expanded in x around -inf 27.2%

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

      1. mul-1-neg 27.2%

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

      2. distribute-neg-frac2 27.2%

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

   7. Simplified 27.2%

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

   if 1.54999999999999999e-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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around inf 53.3%

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

      1. *-commutative 53.3%

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

   7. Simplified 53.3%

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

      1. unpow2 53.3%

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

      2. unpow2 53.3%

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

      3. times-frac 53.4%

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

   9. Applied egg-rr 53.4%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-150}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.3% 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 79.9%

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

   1. add-sqr-sqrt 79.1%

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

   2. pow2 79.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}\right)}^{2}} \]

4. Applied egg-rr 79.1%

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

5. Taylor expanded in x around -inf 16.3%

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

   1. mul-1-neg 16.3%

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

   2. distribute-neg-frac2 16.3%

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

7. Simplified 16.3%

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

8. Final simplification 16.3%

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

Developer target: 78.8% 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 2024072 
(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))))))))