Given's Rotation SVD example

Percentage Accurate: 79.0% → 99.8%

Time: 8.5s

Alternatives: 9

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: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.9995)
   (/ p_m (- x))
   (sqrt (fma (/ 0.5 (hypot x (* p_m 2.0))) x 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)))) <= -0.9995) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(fma((0.5 / hypot(x, (p_m * 2.0))), x, 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)))) <= -0.9995)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(fma(Float64(0.5 / hypot(x, Float64(p_m * 2.0))), x, 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], -0.9995], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * x + 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)))) < -0.99950000000000006

   1. Initial program 10.0%

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

      1. expm1-log1p-u 10.0%

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

      2. expm1-undefine 10.0%

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

      3. +-commutative 10.0%

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

      4. add-sqr-sqrt 10.0%

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

      5. hypot-define 10.0%

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

      6. associate-*l* 10.0%

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

      7. sqrt-prod 10.0%

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

      8. metadata-eval 10.0%

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

      9. sqrt-unprod 5.5%

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

      10. add-sqr-sqrt 10.0%

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

   4. Applied egg-rr 10.0%

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

      1. sub-neg 10.0%

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

      2. metadata-eval 10.0%

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

      3. +-commutative 10.0%

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

      4. log1p-undefine 10.0%

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

      5. rem-exp-log 10.0%

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

      6. associate-+r+ 10.0%

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

      7. metadata-eval 10.0%

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

   6. Simplified 10.0%

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

   7. Taylor expanded in x around -inf 56.0%

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

      1. associate-*r/ 56.0%

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

      2. *-commutative 56.0%

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

      3. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   10. Taylor expanded in p around -inf 53.5%

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

       1. mul-1-neg 53.5%

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

       2. distribute-neg-frac2 53.5%

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

   12. Simplified 53.5%

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

   if -0.99950000000000006 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.7%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-undefine 99.2%

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

      3. +-commutative 99.2%

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

      4. add-sqr-sqrt 99.2%

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

      5. hypot-define 99.2%

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

      6. associate-*l* 99.2%

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

      7. sqrt-prod 99.4%

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

      8. metadata-eval 99.4%

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

      9. sqrt-unprod 41.3%

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

      10. add-sqr-sqrt 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. +-commutative 99.4%

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

      4. log1p-undefine 99.9%

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

      5. rem-exp-log 99.9%

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

      6. associate-+r+ 99.9%

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

      7. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-/r/ 99.9%

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

      4. fma-define 99.9%

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

      5. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 87.6%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.99950000000000006

   1. Initial program 10.0%

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

      1. expm1-log1p-u 10.0%

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

      2. expm1-undefine 10.0%

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

      3. +-commutative 10.0%

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

      4. add-sqr-sqrt 10.0%

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

      5. hypot-define 10.0%

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

      6. associate-*l* 10.0%

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

      7. sqrt-prod 10.0%

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

      8. metadata-eval 10.0%

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

      9. sqrt-unprod 5.5%

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

      10. add-sqr-sqrt 10.0%

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

   4. Applied egg-rr 10.0%

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

      1. sub-neg 10.0%

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

      2. metadata-eval 10.0%

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

      3. +-commutative 10.0%

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

      4. log1p-undefine 10.0%

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

      5. rem-exp-log 10.0%

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

      6. associate-+r+ 10.0%

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

      7. metadata-eval 10.0%

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

   6. Simplified 10.0%

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

   7. Taylor expanded in x around -inf 56.0%

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

      1. associate-*r/ 56.0%

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

      2. *-commutative 56.0%

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

      3. associate-/l* 56.0%

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

   9. Simplified 56.0%

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

   10. Taylor expanded in p around -inf 53.5%

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

       1. mul-1-neg 53.5%

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

       2. distribute-neg-frac2 53.5%

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

   12. Simplified 53.5%

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

   if -0.99950000000000006 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.7%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-undefine 99.2%

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

      3. +-commutative 99.2%

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

      4. add-sqr-sqrt 99.2%

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

      5. hypot-define 99.2%

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

      6. associate-*l* 99.2%

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

      7. sqrt-prod 99.4%

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

      8. metadata-eval 99.4%

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

      9. sqrt-unprod 41.3%

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

      10. add-sqr-sqrt 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. +-commutative 99.4%

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

      4. log1p-undefine 99.9%

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

      5. rem-exp-log 99.9%

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

      6. associate-+r+ 99.9%

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

      7. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

      2. associate-/r/ 99.9%

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

      3. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 87.6%

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

Alternative 3: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -3.5e-55)
   (/ 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 <= -3.5e-55) {
		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 <= -3.5e-55) {
		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 <= -3.5e-55:
		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 <= -3.5e-55)
		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 <= -3.5e-55)
		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, -3.5e-55], 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 < -3.50000000000000025e-55

   1. Initial program 43.9%

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

      1. expm1-log1p-u 43.9%

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

      2. expm1-undefine 43.9%

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

      3. +-commutative 43.9%

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

      4. add-sqr-sqrt 43.9%

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

      5. hypot-define 43.9%

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

      6. associate-*l* 43.9%

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

      7. sqrt-prod 44.3%

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

      8. metadata-eval 44.3%

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

      9. sqrt-unprod 20.1%

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

      10. add-sqr-sqrt 44.3%

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

   4. Applied egg-rr 44.3%

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

      1. sub-neg 44.3%

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

      2. metadata-eval 44.3%

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

      3. +-commutative 44.3%

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

      4. log1p-undefine 44.3%

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

      5. rem-exp-log 44.3%

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

      6. associate-+r+ 44.3%

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

      7. metadata-eval 44.3%

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

   6. Simplified 44.3%

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

   7. Taylor expanded in x around -inf 41.4%

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

      1. associate-*r/ 41.3%

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

      2. *-commutative 41.3%

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

      3. associate-/l* 41.4%

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

   9. Simplified 41.4%

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

   10. Taylor expanded in p around -inf 35.7%

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

       1. mul-1-neg 35.7%

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

       2. distribute-neg-frac2 35.7%

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

   12. Simplified 35.7%

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

   if -3.50000000000000025e-55 < x

   1. Initial program 91.8%

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

      1. add-sqr-sqrt 91.8%

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

      2. hypot-define 91.8%

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

      3. associate-*l* 91.8%

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

      4. sqrt-prod 91.8%

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

      5. metadata-eval 91.8%

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

      6. sqrt-unprod 37.8%

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

      7. add-sqr-sqrt 91.8%

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

   4. Applied egg-rr 91.8%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 3.8 \cdot 10^{-261}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.4 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.08 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.4 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 7 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p\_m \leq 1.22 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p\_m}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 3.8e-261)
     1.0
     (if (<= p_m 2.4e-113)
       t_0
       (if (<= p_m 1.08e-92)
         1.0
         (if (<= p_m 1.4e-34)
           t_0
           (if (<= p_m 7e+17)
             (sqrt 0.5)
             (if (<= p_m 1.22e+39)
               t_0
               (sqrt (+ 0.5 (/ (* x 0.25) p_m)))))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 3.8e-261) {
		tmp = 1.0;
	} else if (p_m <= 2.4e-113) {
		tmp = t_0;
	} else if (p_m <= 1.08e-92) {
		tmp = 1.0;
	} else if (p_m <= 1.4e-34) {
		tmp = t_0;
	} else if (p_m <= 7e+17) {
		tmp = sqrt(0.5);
	} else if (p_m <= 1.22e+39) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 3.8d-261) then
        tmp = 1.0d0
    else if (p_m <= 2.4d-113) then
        tmp = t_0
    else if (p_m <= 1.08d-92) then
        tmp = 1.0d0
    else if (p_m <= 1.4d-34) then
        tmp = t_0
    else if (p_m <= 7d+17) then
        tmp = sqrt(0.5d0)
    else if (p_m <= 1.22d+39) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 + ((x * 0.25d0) / p_m)))
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 3.8e-261) {
		tmp = 1.0;
	} else if (p_m <= 2.4e-113) {
		tmp = t_0;
	} else if (p_m <= 1.08e-92) {
		tmp = 1.0;
	} else if (p_m <= 1.4e-34) {
		tmp = t_0;
	} else if (p_m <= 7e+17) {
		tmp = Math.sqrt(0.5);
	} else if (p_m <= 1.22e+39) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.25) / p_m)));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 3.8e-261:
		tmp = 1.0
	elif p_m <= 2.4e-113:
		tmp = t_0
	elif p_m <= 1.08e-92:
		tmp = 1.0
	elif p_m <= 1.4e-34:
		tmp = t_0
	elif p_m <= 7e+17:
		tmp = math.sqrt(0.5)
	elif p_m <= 1.22e+39:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 + ((x * 0.25) / p_m)))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 3.8e-261)
		tmp = 1.0;
	elseif (p_m <= 2.4e-113)
		tmp = t_0;
	elseif (p_m <= 1.08e-92)
		tmp = 1.0;
	elseif (p_m <= 1.4e-34)
		tmp = t_0;
	elseif (p_m <= 7e+17)
		tmp = sqrt(0.5);
	elseif (p_m <= 1.22e+39)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.25) / p_m)));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 3.8e-261)
		tmp = 1.0;
	elseif (p_m <= 2.4e-113)
		tmp = t_0;
	elseif (p_m <= 1.08e-92)
		tmp = 1.0;
	elseif (p_m <= 1.4e-34)
		tmp = t_0;
	elseif (p_m <= 7e+17)
		tmp = sqrt(0.5);
	elseif (p_m <= 1.22e+39)
		tmp = t_0;
	else
		tmp = sqrt((0.5 + ((x * 0.25) / p_m)));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 3.8e-261], 1.0, If[LessEqual[p$95$m, 2.4e-113], t$95$0, If[LessEqual[p$95$m, 1.08e-92], 1.0, If[LessEqual[p$95$m, 1.4e-34], t$95$0, If[LessEqual[p$95$m, 7e+17], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p$95$m, 1.22e+39], t$95$0, N[Sqrt[N[(0.5 + N[(N[(x * 0.25), $MachinePrecision] / p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 3.8e-261 or 2.40000000000000012e-113 < p < 1.08e-92

   1. Initial program 78.0%

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

   3. Taylor expanded in x around inf 40.6%

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

   if 3.8e-261 < p < 2.40000000000000012e-113 or 1.08e-92 < p < 1.39999999999999998e-34 or 7e17 < p < 1.22e39

   1. Initial program 33.8%

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

      1. expm1-log1p-u 33.4%

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

      2. expm1-undefine 33.4%

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

      3. +-commutative 33.4%

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

      4. add-sqr-sqrt 33.4%

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

      5. hypot-define 33.4%

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

      6. associate-*l* 33.4%

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

      7. sqrt-prod 33.4%

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

      8. metadata-eval 33.4%

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

      9. sqrt-unprod 33.4%

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

      10. add-sqr-sqrt 33.4%

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

   4. Applied egg-rr 33.4%

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

      1. sub-neg 33.4%

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

      2. metadata-eval 33.4%

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

      3. +-commutative 33.4%

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

      4. log1p-undefine 33.8%

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

      5. rem-exp-log 33.8%

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

      6. associate-+r+ 33.8%

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

      7. metadata-eval 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in x around -inf 43.7%

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

      1. associate-*r/ 43.7%

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

      2. *-commutative 43.7%

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

      3. associate-/l* 43.7%

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

   9. Simplified 43.7%

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

   10. Taylor expanded in p around -inf 73.1%

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

       1. mul-1-neg 73.1%

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

       2. distribute-neg-frac2 73.1%

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

   12. Simplified 73.1%

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

   if 1.39999999999999998e-34 < p < 7e17

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0 54.6%

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

   if 1.22e39 < p

   1. Initial program 98.0%

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

      1. expm1-log1p-u 98.0%

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

      2. expm1-undefine 97.9%

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

      3. +-commutative 97.9%

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

      4. add-sqr-sqrt 97.9%

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

      5. hypot-define 97.9%

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

      6. associate-*l* 97.9%

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

      7. sqrt-prod 97.9%

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

      8. metadata-eval 97.9%

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

      9. sqrt-unprod 97.9%

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

      10. add-sqr-sqrt 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. +-commutative 97.9%

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

      4. log1p-undefine 98.0%

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

      5. rem-exp-log 98.0%

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

      6. associate-+r+ 98.0%

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

      7. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   8. Applied egg-rr 98.0%

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

      1. *-lft-identity 98.0%

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

      2. associate-/r/ 98.0%

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

      3. *-commutative 98.0%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in x around 0 94.2%

       \[\leadsto \sqrt{0.5 + \color{blue}{0.25 \cdot \frac{x}{p}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 94.2%

          \[\leadsto \sqrt{0.5 + \color{blue}{\frac{0.25 \cdot x}{p}}} \]

       2. *-commutative 94.2%

          \[\leadsto \sqrt{0.5 + \frac{\color{blue}{x \cdot 0.25}}{p}} \]

   13. Simplified 94.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3.8 \cdot 10^{-261}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.4 \cdot 10^{-113}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.08 \cdot 10^{-92}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 7 \cdot 10^{+17}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 1.22 \cdot 10^{+39}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.25}{p}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 4.2 \cdot 10^{-261}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.05 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.4 \cdot 10^{-34} \lor \neg \left(p\_m \leq 5 \cdot 10^{+17}\right) \land p\_m \leq 1.75 \cdot 10^{+32}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 4.2e-261)
     1.0
     (if (<= p_m 2.05e-113)
       t_0
       (if (<= p_m 5.8e-93)
         1.0
         (if (or (<= p_m 1.4e-34) (and (not (<= p_m 5e+17)) (<= p_m 1.75e+32)))
           t_0
           (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 4.2e-261) {
		tmp = 1.0;
	} else if (p_m <= 2.05e-113) {
		tmp = t_0;
	} else if (p_m <= 5.8e-93) {
		tmp = 1.0;
	} else if ((p_m <= 1.4e-34) || (!(p_m <= 5e+17) && (p_m <= 1.75e+32))) {
		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 = p_m / -x
    if (p_m <= 4.2d-261) then
        tmp = 1.0d0
    else if (p_m <= 2.05d-113) then
        tmp = t_0
    else if (p_m <= 5.8d-93) then
        tmp = 1.0d0
    else if ((p_m <= 1.4d-34) .or. (.not. (p_m <= 5d+17)) .and. (p_m <= 1.75d+32)) 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 = p_m / -x;
	double tmp;
	if (p_m <= 4.2e-261) {
		tmp = 1.0;
	} else if (p_m <= 2.05e-113) {
		tmp = t_0;
	} else if (p_m <= 5.8e-93) {
		tmp = 1.0;
	} else if ((p_m <= 1.4e-34) || (!(p_m <= 5e+17) && (p_m <= 1.75e+32))) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 4.2e-261:
		tmp = 1.0
	elif p_m <= 2.05e-113:
		tmp = t_0
	elif p_m <= 5.8e-93:
		tmp = 1.0
	elif (p_m <= 1.4e-34) or (not (p_m <= 5e+17) and (p_m <= 1.75e+32)):
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 4.2e-261)
		tmp = 1.0;
	elseif (p_m <= 2.05e-113)
		tmp = t_0;
	elseif (p_m <= 5.8e-93)
		tmp = 1.0;
	elseif ((p_m <= 1.4e-34) || (!(p_m <= 5e+17) && (p_m <= 1.75e+32)))
		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 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 4.2e-261)
		tmp = 1.0;
	elseif (p_m <= 2.05e-113)
		tmp = t_0;
	elseif (p_m <= 5.8e-93)
		tmp = 1.0;
	elseif ((p_m <= 1.4e-34) || (~((p_m <= 5e+17)) && (p_m <= 1.75e+32)))
		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[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 4.2e-261], 1.0, If[LessEqual[p$95$m, 2.05e-113], t$95$0, If[LessEqual[p$95$m, 5.8e-93], 1.0, If[Or[LessEqual[p$95$m, 1.4e-34], And[N[Not[LessEqual[p$95$m, 5e+17]], $MachinePrecision], LessEqual[p$95$m, 1.75e+32]]], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.19999999999999991e-261 or 2.05e-113 < p < 5.7999999999999997e-93

   1. Initial program 78.0%

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

   3. Taylor expanded in x around inf 40.6%

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

   if 4.19999999999999991e-261 < p < 2.05e-113 or 5.7999999999999997e-93 < p < 1.39999999999999998e-34 or 5e17 < p < 1.75e32

   1. Initial program 32.7%

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

      1. expm1-log1p-u 32.3%

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

      2. expm1-undefine 32.3%

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

      3. +-commutative 32.3%

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

      4. add-sqr-sqrt 32.3%

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

      5. hypot-define 32.3%

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

      6. associate-*l* 32.3%

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

      7. sqrt-prod 32.3%

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

      8. metadata-eval 32.3%

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

      9. sqrt-unprod 32.3%

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

      10. add-sqr-sqrt 32.3%

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

   4. Applied egg-rr 32.3%

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

      1. sub-neg 32.3%

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

      2. metadata-eval 32.3%

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

      3. +-commutative 32.3%

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

      4. log1p-undefine 32.7%

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

      5. rem-exp-log 32.7%

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

      6. associate-+r+ 32.7%

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

      7. metadata-eval 32.7%

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

   6. Simplified 32.7%

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

   7. Taylor expanded in x around -inf 43.1%

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

      1. associate-*r/ 43.1%

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

      2. *-commutative 43.1%

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

      3. associate-/l* 43.1%

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

   9. Simplified 43.1%

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

   10. Taylor expanded in p around -inf 74.4%

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

       1. mul-1-neg 74.4%

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

       2. distribute-neg-frac2 74.4%

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

   12. Simplified 74.4%

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

   if 1.39999999999999998e-34 < p < 5e17 or 1.75e32 < p

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 84.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.2 \cdot 10^{-261}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.05 \cdot 10^{-113}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 5.8 \cdot 10^{-93}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-34} \lor \neg \left(p \leq 5 \cdot 10^{+17}\right) \land p \leq 1.75 \cdot 10^{+32}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.4 \cdot 10^{-34} \lor \neg \left(p\_m \leq 7 \cdot 10^{+17}\right) \land p\_m \leq 1.75 \cdot 10^{+32}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (or (<= p_m 1.4e-34) (and (not (<= p_m 7e+17)) (<= p_m 1.75e+32)))
   (/ p_m (- x))
   (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((p_m <= 1.4e-34) || (!(p_m <= 7e+17) && (p_m <= 1.75e+32))) {
		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 <= 1.4d-34) .or. (.not. (p_m <= 7d+17)) .and. (p_m <= 1.75d+32)) 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 <= 1.4e-34) || (!(p_m <= 7e+17) && (p_m <= 1.75e+32))) {
		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 <= 1.4e-34) or (not (p_m <= 7e+17) and (p_m <= 1.75e+32)):
		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 <= 1.4e-34) || (!(p_m <= 7e+17) && (p_m <= 1.75e+32)))
		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 <= 1.4e-34) || (~((p_m <= 7e+17)) && (p_m <= 1.75e+32)))
		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[Or[LessEqual[p$95$m, 1.4e-34], And[N[Not[LessEqual[p$95$m, 7e+17]], $MachinePrecision], LessEqual[p$95$m, 1.75e+32]]], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.39999999999999998e-34 or 7e17 < p < 1.75e32

   1. Initial program 69.7%

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

      1. expm1-log1p-u 69.2%

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

      2. expm1-undefine 69.3%

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

      3. +-commutative 69.3%

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

      4. add-sqr-sqrt 69.3%

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

      5. hypot-define 69.3%

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

      6. associate-*l* 69.3%

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

      7. sqrt-prod 69.4%

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

      8. metadata-eval 69.4%

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

      9. sqrt-unprod 10.6%

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

      10. add-sqr-sqrt 69.4%

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

   4. Applied egg-rr 69.4%

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

      1. sub-neg 69.4%

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

      2. metadata-eval 69.4%

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

      3. +-commutative 69.4%

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

      4. log1p-undefine 69.9%

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

      5. rem-exp-log 69.9%

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

      6. associate-+r+ 69.9%

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

      7. metadata-eval 69.9%

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

   6. Simplified 69.9%

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

   7. Taylor expanded in x around -inf 21.1%

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

      1. associate-*r/ 21.1%

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

      2. *-commutative 21.1%

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

      3. associate-/l* 21.1%

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

   9. Simplified 21.1%

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

   10. Taylor expanded in p around -inf 19.6%

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

       1. mul-1-neg 19.6%

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

       2. distribute-neg-frac2 19.6%

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

   12. Simplified 19.6%

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

   if 1.39999999999999998e-34 < p < 7e17 or 1.75e32 < p

   1. Initial program 94.1%

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

   3. Taylor expanded in x around 0 84.0%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.4 \cdot 10^{-34} \lor \neg \left(p \leq 7 \cdot 10^{+17}\right) \land p \leq 1.75 \cdot 10^{+32}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.1% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -2e-311) (/ p_m (- x)) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2e-311) {
		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 <= (-2d-311)) 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 <= -2e-311) {
		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 <= -2e-311:
		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 <= -2e-311)
		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 <= -2e-311)
		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, -2e-311], N[(p$95$m / (-x)), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999e-311

   1. Initial program 51.4%

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

      1. expm1-log1p-u 51.4%

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

      2. expm1-undefine 51.4%

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

      3. +-commutative 51.4%

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

      4. add-sqr-sqrt 51.4%

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

      5. hypot-define 51.4%

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

      6. associate-*l* 51.4%

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

      7. sqrt-prod 51.7%

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

      8. metadata-eval 51.7%

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

      9. sqrt-unprod 22.6%

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

      10. add-sqr-sqrt 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. sub-neg 51.7%

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

      2. metadata-eval 51.7%

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

      3. +-commutative 51.7%

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

      4. log1p-undefine 51.7%

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

      5. rem-exp-log 51.7%

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

      6. associate-+r+ 51.7%

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

      7. metadata-eval 51.7%

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

   6. Simplified 51.7%

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

   7. Taylor expanded in x around -inf 32.3%

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

      1. associate-*r/ 32.2%

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

      2. *-commutative 32.2%

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

      3. associate-/l* 32.3%

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

   9. Simplified 32.3%

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

   10. Taylor expanded in p around -inf 29.9%

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

       1. mul-1-neg 29.9%

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

       2. distribute-neg-frac2 29.9%

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

   12. Simplified 29.9%

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

   if -1.9999999999999e-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. Step-by-step derivation

      1. expm1-log1p-u 99.2%

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

      2. expm1-undefine 99.2%

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

      3. +-commutative 99.2%

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

      4. add-sqr-sqrt 99.2%

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

      5. hypot-define 99.2%

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

      6. associate-*l* 99.2%

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

      7. sqrt-prod 99.2%

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

      8. metadata-eval 99.2%

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

      9. sqrt-unprod 40.8%

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

      10. add-sqr-sqrt 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. sub-neg 99.2%

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

      2. metadata-eval 99.2%

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

      3. +-commutative 99.2%

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

      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%

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

   7. Taylor expanded in x around -inf 4.9%

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

      1. associate-*r/ 4.9%

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

      2. *-commutative 4.9%

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

      3. associate-/l* 4.9%

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

   9. Simplified 4.9%

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

   10. Taylor expanded in p around 0 3.2%

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

Alternative 8: 6.6% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. expm1-log1p-u 75.5%

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

   2. expm1-undefine 75.5%

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

   3. +-commutative 75.5%

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

   4. add-sqr-sqrt 75.5%

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

   5. hypot-define 75.5%

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

   6. associate-*l* 75.5%

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

   7. sqrt-prod 75.7%

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

   8. metadata-eval 75.7%

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

   9. sqrt-unprod 31.8%

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

   10. add-sqr-sqrt 75.7%

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

4. Applied egg-rr 75.7%

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

   1. sub-neg 75.7%

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

   2. metadata-eval 75.7%

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

   3. +-commutative 75.7%

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

   4. log1p-undefine 76.0%

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

   5. rem-exp-log 76.0%

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

   6. associate-+r+ 76.0%

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

   7. metadata-eval 76.0%

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

6. Simplified 76.0%

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

7. Taylor expanded in x around -inf 18.5%

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

   1. associate-*r/ 18.5%

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

   2. *-commutative 18.5%

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

   3. associate-/l* 18.5%

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

9. Simplified 18.5%

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

10. Taylor expanded in p around 0 16.9%

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

Alternative 9: 6.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ 0 \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 0.0)

C

p_m = fabs(p);
double code(double p_m, double x) {
	return 0.0;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	return 0.0;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 0.0

Julia

p_m = abs(p)
function code(p_m, x)
	return 0.0
end

MATLAB

p_m = abs(p);
function tmp = code(p_m, x)
	tmp = 0.0;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 0.0

Derivation

1. Initial program 75.9%

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

3. Taylor expanded in x around -inf 4.8%

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

   1. metadata-eval 4.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{0}} \]

   2. metadata-eval 4.8%

      \[\leadsto \sqrt{\color{blue}{0}} \]

   3. metadata-eval 4.8%

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

   4. pow1/2 4.8%

      \[\leadsto \color{blue}{{\left(1 + -1\right)}^{0.5}} \]

   5. metadata-eval 4.8%

      \[\leadsto {\color{blue}{0}}^{0.5} \]

   6. metadata-eval 4.8%

      \[\leadsto \color{blue}{0} \]

5. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{0} \]
6. Add Preprocessing

Developer target: 79.0% 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 2024111 
(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))))))))