Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.1%

Time: 15.1s

Alternatives: 8

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: 79.4% 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: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{0.125 \cdot \left(\frac{-12}{p\_m} \cdot \frac{{p\_m}^{4}}{-{x}^{2}}\right) - p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}\right)\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.5)
   (/ (- (* 0.125 (* (/ -12.0 p_m) (/ (pow p_m 4.0) (- (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)))) <= -0.5) {
		tmp = ((0.125 * ((-12.0 / p_m) * (pow(p_m, 4.0) / -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)))) <= -0.5) {
		tmp = ((0.125 * ((-12.0 / p_m) * (Math.pow(p_m, 4.0) / -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)))) <= -0.5:
		tmp = ((0.125 * ((-12.0 / p_m) * (math.pow(p_m, 4.0) / -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)))) <= -0.5)
		tmp = Float64(Float64(Float64(0.125 * Float64(Float64(-12.0 / p_m) * Float64((p_m ^ 4.0) / Float64(-(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)))) <= -0.5)
		tmp = ((0.125 * ((-12.0 / p_m) * ((p_m ^ 4.0) / -(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], -0.5], N[(N[(N[(0.125 * N[(N[(-12.0 / p$95$m), $MachinePrecision] * N[(N[Power[p$95$m, 4.0], $MachinePrecision] / (-N[Power[x, 2.0], $MachinePrecision])), $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)))) < -0.5

   1. Initial program 19.5%

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

      1. sqrt-prod 19.5%

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

      2. *-commutative 19.5%

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

      3. +-commutative 19.5%

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

      4. add-sqr-sqrt 19.5%

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

      5. hypot-define 19.5%

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

      6. associate-*l* 19.5%

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

      7. sqrt-prod 19.5%

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

      8. metadata-eval 19.5%

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

      9. sqrt-unprod 5.8%

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

      10. add-sqr-sqrt 19.5%

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

   4. Applied egg-rr 19.5%

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

      1. sqrt-unprod 19.5%

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

      2. metadata-eval 19.5%

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

      3. *-commutative 19.5%

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

      4. associate-+r+ 19.5%

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

      5. *-commutative 19.5%

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

      6. add-cbrt-cube 19.5%

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

      7. add-sqr-sqrt 19.5%

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

      8. pow1 19.5%

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

      9. pow1/2 19.5%

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

      10. pow-prod-up 19.5%

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

   6. Applied egg-rr 19.5%

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

   7. Taylor expanded in x around -inf 50.9%

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

      1. mul-1-neg 50.9%

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

      2. distribute-neg-frac2 50.9%

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

      3. distribute-rgt-out 50.9%

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

      4. *-commutative 50.9%

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

      5. times-frac 60.4%

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

      6. metadata-eval 60.4%

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

   9. Simplified 60.4%

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

   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. 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 43.2%

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

      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%

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

      8. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{0.125 \cdot \left(\frac{-12}{p} \cdot \frac{{p}^{4}}{-{x}^{2}}\right) - p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(-1 + \left(2 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.7× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ 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)))) <= -1.0) {
		tmp = 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)))) <= -1.0) {
		tmp = 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)))) <= -1.0:
		tmp = 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)))) <= -1.0)
		tmp = Float64(p_m / Float64(-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)))) <= -1.0)
		tmp = 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], -1.0], N[(p$95$m / (-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)))) < -1

   1. Initial program 18.1%

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

   3. Taylor expanded in x around -inf 61.3%

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

      1. mul-1-neg 61.3%

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

   5. Simplified 61.3%

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

      1. distribute-neg-frac 61.3%

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

      2. sqrt-unprod 62.2%

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

      3. metadata-eval 62.2%

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

      4. metadata-eval 62.2%

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

      5. *-rgt-identity 62.2%

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

   7. Applied egg-rr 62.2%

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

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

   1. Initial program 99.6%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.1%

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

      3. +-commutative 99.1%

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

      4. add-sqr-sqrt 99.1%

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

      5. hypot-define 99.1%

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

      6. associate-*l* 99.1%

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

      7. sqrt-prod 99.1%

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

      8. metadata-eval 99.1%

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

      9. sqrt-unprod 42.8%

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

      10. add-sqr-sqrt 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. log1p-undefine 99.6%

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

      5. rem-exp-log 99.6%

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

      6. associate-+r+ 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 90.4%

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

Alternative 3: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}{x}}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5.4e-40)
   (/ p_m (- x))
   (sqrt (+ 0.5 (/ 0.5 (/ (hypot x (* p_m 2.0)) x))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5.4e-40) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + (0.5 / (hypot(x, (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 <= -5.4e-40) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 / (Math.hypot(x, (p_m * 2.0)) / x))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5.4e-40:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + (0.5 / (math.hypot(x, (p_m * 2.0)) / x))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5.4e-40)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 / Float64(hypot(x, 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 <= -5.4e-40)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + (0.5 / (hypot(x, (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, -5.4e-40], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 / N[(N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.4e-40

   1. Initial program 47.4%

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

   3. Taylor expanded in x around -inf 42.1%

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

      1. mul-1-neg 42.1%

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

   5. Simplified 42.1%

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

      1. distribute-neg-frac 42.1%

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

      2. sqrt-unprod 42.7%

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

      3. metadata-eval 42.7%

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

      4. metadata-eval 42.7%

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

      5. *-rgt-identity 42.7%

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

   7. Applied egg-rr 42.7%

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

   if -5.4e-40 < x

   1. Initial program 93.4%

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

      1. +-commutative 93.4%

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

      2. distribute-rgt-in 93.4%

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

      3. +-commutative 93.4%

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

      4. add-sqr-sqrt 93.4%

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

      5. hypot-define 93.4%

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

      6. associate-*l* 93.4%

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

      7. sqrt-prod 93.4%

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

      8. metadata-eval 93.4%

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

      9. sqrt-unprod 39.8%

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

      10. add-sqr-sqrt 93.4%

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

      11. metadata-eval 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. *-commutative 93.4%

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

      2. clear-num 93.4%

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

      3. un-div-inv 93.4%

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

   6. Applied egg-rr 93.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4 \cdot 10^{-40}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1.6e-38)
   (/ p_m (- x))
   (sqrt (+ 0.5 (* 0.5 (/ x (hypot x (* p_m 2.0))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.6e-38) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1.6e-38) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + (0.5 * (x / Math.hypot(x, (p_m * 2.0))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.6e-38:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + (0.5 * (x / math.hypot(x, (p_m * 2.0))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.6e-38)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(0.5 * Float64(x / hypot(x, Float64(p_m * 2.0))))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.6e-38)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + (0.5 * (x / hypot(x, (p_m * 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.6e-38], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(0.5 * N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999989e-38

   1. Initial program 47.4%

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

   3. Taylor expanded in x around -inf 42.1%

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

      1. mul-1-neg 42.1%

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

   5. Simplified 42.1%

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

      1. distribute-neg-frac 42.1%

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

      2. sqrt-unprod 42.7%

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

      3. metadata-eval 42.7%

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

      4. metadata-eval 42.7%

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

      5. *-rgt-identity 42.7%

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

   7. Applied egg-rr 42.7%

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

   if -1.59999999999999989e-38 < x

   1. Initial program 93.4%

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

      1. +-commutative 93.4%

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

      2. distribute-rgt-in 93.4%

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

      3. +-commutative 93.4%

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

      4. add-sqr-sqrt 93.4%

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

      5. hypot-define 93.4%

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

      6. associate-*l* 93.4%

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

      7. sqrt-prod 93.4%

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

      8. metadata-eval 93.4%

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

      9. sqrt-unprod 39.8%

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

      10. add-sqr-sqrt 93.4%

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

      11. metadata-eval 93.4%

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

   4. Applied egg-rr 93.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.35 \cdot 10^{-294}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 2.75 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.35e-294) (/ p_m (- x)) (if (<= p_m 2.75e-64) 1.0 (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.35e-294) {
		tmp = p_m / -x;
	} else if (p_m <= 2.75e-64) {
		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 <= 1.35d-294) then
        tmp = p_m / -x
    else if (p_m <= 2.75d-64) 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 <= 1.35e-294) {
		tmp = p_m / -x;
	} else if (p_m <= 2.75e-64) {
		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 <= 1.35e-294:
		tmp = p_m / -x
	elif p_m <= 2.75e-64:
		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 <= 1.35e-294)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 2.75e-64)
		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 <= 1.35e-294)
		tmp = p_m / -x;
	elseif (p_m <= 2.75e-64)
		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, 1.35e-294], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 2.75e-64], 1.0, N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.35000000000000005e-294

   1. Initial program 81.3%

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

   3. Taylor expanded in x around -inf 10.4%

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

      1. mul-1-neg 10.4%

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

   5. Simplified 10.4%

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

      1. distribute-neg-frac 10.4%

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

      2. sqrt-unprod 10.4%

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

      3. metadata-eval 10.4%

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

      4. metadata-eval 10.4%

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

      5. *-rgt-identity 10.4%

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

   7. Applied egg-rr 10.4%

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

   if 1.35000000000000005e-294 < p < 2.7499999999999999e-64

   1. Initial program 69.3%

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

   3. Taylor expanded in x around inf 60.4%

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

   if 2.7499999999999999e-64 < p

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0 78.0%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.35 \cdot 10^{-294}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 2.75 \cdot 10^{-64}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.65 \cdot 10^{-100}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.65e-100) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.65e-100) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.65d-100) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.65e-100) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.65e-100:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.65e-100)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.65e-100)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.65e-100], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.64999999999999998e-100

   1. Initial program 78.2%

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

   3. Taylor expanded in x around -inf 17.1%

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

      1. mul-1-neg 17.1%

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

   5. Simplified 17.1%

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

      1. distribute-neg-frac 17.1%

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

      2. sqrt-unprod 17.2%

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

      3. metadata-eval 17.2%

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

      4. metadata-eval 17.2%

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

      5. *-rgt-identity 17.2%

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

   7. Applied egg-rr 17.2%

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

   if 1.64999999999999998e-100 < p

   1. Initial program 82.9%

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

   3. Taylor expanded in x around 0 74.2%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.65 \cdot 10^{-100}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 28.7% accurate, 23.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -9.999999999999969e-311

   1. Initial program 59.1%

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

   3. Taylor expanded in x around -inf 31.9%

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

      1. mul-1-neg 31.9%

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

   5. Simplified 31.9%

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

      1. distribute-neg-frac 31.9%

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

      2. sqrt-unprod 32.3%

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

      3. metadata-eval 32.3%

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

      4. metadata-eval 32.3%

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

      5. *-rgt-identity 32.3%

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

   7. Applied egg-rr 32.3%

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

   if -9.999999999999969e-311 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 4.2%

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

      1. mul-1-neg 4.2%

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

   5. Simplified 4.2%

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

      1. neg-sub0 4.2%

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

      2. sub-neg 4.2%

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

      3. add-sqr-sqrt 3.5%

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

      4. sqrt-unprod 5.3%

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

      5. sqr-neg 5.3%

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

      6. sqrt-unprod 2.1%

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

      7. add-sqr-sqrt 3.0%

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

      8. sqrt-unprod 3.0%

         \[\leadsto 0 + \frac{p \cdot \color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

      9. metadata-eval 3.0%

         \[\leadsto 0 + \frac{p \cdot \sqrt{\color{blue}{1}}}{x} \]

      10. metadata-eval 3.0%

          \[\leadsto 0 + \frac{p \cdot \color{blue}{1}}{x} \]

      11. *-rgt-identity 3.0%

          \[\leadsto 0 + \frac{\color{blue}{p}}{x} \]

   7. Applied egg-rr 3.0%

      \[\leadsto \color{blue}{0 + \frac{p}{x}} \]
   8. Step-by-step derivation

      1. +-lft-identity 3.0%

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

   9. Simplified 3.0%

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

4. Final simplification 17.6%

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

Alternative 8: 6.5% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in x around -inf 18.0%

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

   1. mul-1-neg 18.0%

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

5. Simplified 18.0%

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

   1. neg-sub0 18.0%

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

   2. sub-neg 18.0%

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

   3. add-sqr-sqrt 16.7%

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

   4. sqrt-unprod 20.7%

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

   5. sqr-neg 20.7%

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

   6. sqrt-unprod 14.9%

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

   7. add-sqr-sqrt 16.0%

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

   8. sqrt-unprod 16.2%

      \[\leadsto 0 + \frac{p \cdot \color{blue}{\sqrt{0.5 \cdot 2}}}{x} \]

   9. metadata-eval 16.2%

      \[\leadsto 0 + \frac{p \cdot \sqrt{\color{blue}{1}}}{x} \]

   10. metadata-eval 16.2%

       \[\leadsto 0 + \frac{p \cdot \color{blue}{1}}{x} \]

   11. *-rgt-identity 16.2%

       \[\leadsto 0 + \frac{\color{blue}{p}}{x} \]

7. Applied egg-rr 16.2%

   \[\leadsto \color{blue}{0 + \frac{p}{x}} \]
8. Step-by-step derivation

   1. +-lft-identity 16.2%

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

9. Simplified 16.2%

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

Developer Target 1: 79.4% 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 2024165 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

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