Given's Rotation SVD example

Percentage Accurate: 79.9% → 99.7%

Time: 8.5s

Alternatives: 4

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 12.7%

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

      1. expm1-log1p-u 12.7%

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

      2. expm1-undefine 12.5%

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

      3. +-commutative 12.5%

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

      4. add-sqr-sqrt 12.5%

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

      5. hypot-define 12.5%

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

      6. associate-*l* 12.5%

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

      7. sqrt-prod 12.5%

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

      8. metadata-eval 12.5%

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

      9. sqrt-unprod 6.8%

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

      10. add-sqr-sqrt 12.5%

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

   4. Applied egg-rr 12.5%

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

      1. sub-neg 12.5%

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

      2. metadata-eval 12.5%

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

      3. +-commutative 12.5%

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

      4. log1p-undefine 12.5%

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

      5. rem-exp-log 12.5%

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

      6. associate-+r+ 12.5%

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

      7. metadata-eval 12.5%

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

   6. Simplified 12.5%

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

      1. add-cbrt-cube 12.5%

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

      2. add-sqr-sqrt 12.5%

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

      3. pow1 12.5%

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

      4. pow1/2 12.5%

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

      5. pow-prod-up 12.5%

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

      6. associate-+r+ 12.5%

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

      7. metadata-eval 12.5%

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

      8. distribute-lft-in 12.5%

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

      9. metadata-eval 12.5%

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

      10. metadata-eval 12.5%

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

   8. Applied egg-rr 12.5%

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

   9. Taylor expanded in x around -inf 50.9%

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

       1. neg-mul-1 50.9%

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

       2. distribute-neg-frac2 50.9%

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

   11. Simplified 50.9%

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

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

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. hypot-define 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 51.3%

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

      7. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 87.9%

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

Alternative 2: 68.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 3 \cdot 10^{-219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 3.1 \cdot 10^{-176}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.55 \cdot 10^{-171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.64 \cdot 10^{-90}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 3e-219)
     t_0
     (if (<= p_m 3.1e-176)
       1.0
       (if (<= p_m 1.55e-171) t_0 (if (<= p_m 1.64e-90) 1.0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 3e-219) {
		tmp = t_0;
	} else if (p_m <= 3.1e-176) {
		tmp = 1.0;
	} else if (p_m <= 1.55e-171) {
		tmp = t_0;
	} else if (p_m <= 1.64e-90) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 3d-219) then
        tmp = t_0
    else if (p_m <= 3.1d-176) then
        tmp = 1.0d0
    else if (p_m <= 1.55d-171) then
        tmp = t_0
    else if (p_m <= 1.64d-90) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 3e-219) {
		tmp = t_0;
	} else if (p_m <= 3.1e-176) {
		tmp = 1.0;
	} else if (p_m <= 1.55e-171) {
		tmp = t_0;
	} else if (p_m <= 1.64e-90) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 3e-219:
		tmp = t_0
	elif p_m <= 3.1e-176:
		tmp = 1.0
	elif p_m <= 1.55e-171:
		tmp = t_0
	elif p_m <= 1.64e-90:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 3e-219)
		tmp = t_0;
	elseif (p_m <= 3.1e-176)
		tmp = 1.0;
	elseif (p_m <= 1.55e-171)
		tmp = t_0;
	elseif (p_m <= 1.64e-90)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 3e-219)
		tmp = t_0;
	elseif (p_m <= 3.1e-176)
		tmp = 1.0;
	elseif (p_m <= 1.55e-171)
		tmp = t_0;
	elseif (p_m <= 1.64e-90)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 3e-219], t$95$0, If[LessEqual[p$95$m, 3.1e-176], 1.0, If[LessEqual[p$95$m, 1.55e-171], t$95$0, If[LessEqual[p$95$m, 1.64e-90], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.0000000000000001e-219 or 3.09999999999999992e-176 < p < 1.55e-171

   1. Initial program 72.1%

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

      1. expm1-log1p-u 71.6%

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

      2. expm1-undefine 71.6%

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

      3. +-commutative 71.6%

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

      4. add-sqr-sqrt 71.6%

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

      5. hypot-define 71.6%

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

      6. associate-*l* 71.6%

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

      7. sqrt-prod 71.6%

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

      8. metadata-eval 71.6%

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

      9. sqrt-unprod 8.2%

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

      10. add-sqr-sqrt 71.6%

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

   4. Applied egg-rr 71.6%

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

      1. sub-neg 71.6%

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

      2. metadata-eval 71.6%

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

      3. +-commutative 71.6%

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

      4. log1p-undefine 72.0%

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

      5. rem-exp-log 72.0%

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

      6. associate-+r+ 72.1%

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

      7. metadata-eval 72.1%

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

   6. Simplified 72.1%

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

      1. add-cbrt-cube 72.0%

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

      2. add-sqr-sqrt 72.0%

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

      3. pow1 72.0%

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

      4. pow1/2 72.0%

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

      5. pow-prod-up 72.0%

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

      6. associate-+r+ 72.0%

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

      7. metadata-eval 72.0%

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

      8. distribute-lft-in 72.0%

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

      9. metadata-eval 72.0%

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

      10. metadata-eval 72.0%

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

   8. Applied egg-rr 72.0%

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

   9. Taylor expanded in x around -inf 15.3%

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

       1. neg-mul-1 15.3%

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

       2. distribute-neg-frac2 15.3%

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

   11. Simplified 15.3%

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

   if 3.0000000000000001e-219 < p < 3.09999999999999992e-176 or 1.55e-171 < p < 1.6400000000000001e-90

   1. Initial program 74.9%

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

      1. expm1-log1p-u 73.9%

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

      2. expm1-undefine 73.1%

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

      3. +-commutative 73.1%

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

      4. add-sqr-sqrt 73.1%

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

      5. hypot-define 73.1%

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

      6. associate-*l* 73.1%

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

      7. sqrt-prod 73.1%

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

      8. metadata-eval 73.1%

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

      9. sqrt-unprod 73.1%

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

      10. add-sqr-sqrt 73.1%

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

   4. Applied egg-rr 73.1%

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

      1. sub-neg 73.1%

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

      2. metadata-eval 73.1%

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

      3. +-commutative 73.1%

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

      4. log1p-undefine 74.2%

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

      5. rem-exp-log 74.2%

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

      6. associate-+r+ 74.2%

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

      7. metadata-eval 74.2%

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

   6. Simplified 74.2%

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

      1. add-cbrt-cube 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. pow1 74.2%

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

      4. pow1/2 74.2%

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

      5. pow-prod-up 74.2%

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

      6. associate-+r+ 74.2%

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

      7. metadata-eval 74.2%

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

      8. distribute-lft-in 74.2%

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

      9. metadata-eval 74.2%

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

      10. metadata-eval 74.2%

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

   8. Applied egg-rr 74.2%

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

   9. Taylor expanded in x around inf 70.7%

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

   if 1.6400000000000001e-90 < p

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0 80.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3 \cdot 10^{-219}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 3.1 \cdot 10^{-176}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.55 \cdot 10^{-171}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.64 \cdot 10^{-90}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.0% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -5e-142) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-142) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-142)) then
        tmp = p_m / -x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -5e-142) {
		tmp = p_m / -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5e-142:
		tmp = p_m / -x
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5e-142)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5e-142)
		tmp = p_m / -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5e-142], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000002e-142

   1. Initial program 54.2%

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

      1. expm1-log1p-u 54.2%

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

      2. expm1-undefine 54.0%

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

      3. +-commutative 54.0%

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

      4. add-sqr-sqrt 54.0%

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

      5. hypot-define 54.0%

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

      6. associate-*l* 54.0%

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

      7. sqrt-prod 54.0%

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

      8. metadata-eval 54.0%

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

      9. sqrt-unprod 29.4%

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

      10. add-sqr-sqrt 54.0%

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

   4. Applied egg-rr 54.0%

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

      1. sub-neg 54.0%

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

      2. metadata-eval 54.0%

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

      3. +-commutative 54.0%

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

      4. log1p-undefine 54.0%

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

      5. rem-exp-log 54.0%

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

      6. associate-+r+ 54.0%

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

      7. metadata-eval 54.0%

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

   6. Simplified 54.0%

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

      1. add-cbrt-cube 54.0%

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

      2. add-sqr-sqrt 54.0%

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

      3. pow1 54.0%

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

      4. pow1/2 54.0%

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

      5. pow-prod-up 54.0%

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

      6. associate-+r+ 54.0%

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

      7. metadata-eval 54.0%

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

      8. distribute-lft-in 54.0%

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

      9. metadata-eval 54.0%

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

      10. metadata-eval 54.0%

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

   8. Applied egg-rr 54.0%

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

   9. Taylor expanded in x around -inf 28.3%

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

       1. neg-mul-1 28.3%

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

       2. distribute-neg-frac2 28.3%

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

   11. Simplified 28.3%

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

   if -5.0000000000000002e-142 < 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 49.6%

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

      10. add-sqr-sqrt 99.2%

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

   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. Step-by-step derivation

      1. add-cbrt-cube 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. pow1 99.9%

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

      4. pow1/2 99.9%

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

      5. pow-prod-up 99.9%

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

      6. associate-+r+ 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-lft-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in x around inf 59.1%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-142}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.3% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

   1. expm1-log1p-u 78.1%

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

   2. expm1-undefine 78.0%

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

   3. +-commutative 78.0%

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

   4. add-sqr-sqrt 78.0%

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

   5. hypot-define 78.0%

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

   6. associate-*l* 78.0%

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

   7. sqrt-prod 78.0%

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

   8. metadata-eval 78.0%

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

   9. sqrt-unprod 40.1%

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

   10. add-sqr-sqrt 78.0%

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

4. Applied egg-rr 78.0%

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

   1. sub-neg 78.0%

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

   2. metadata-eval 78.0%

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

   3. +-commutative 78.0%

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

   4. log1p-undefine 78.4%

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

   5. rem-exp-log 78.4%

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

   6. associate-+r+ 78.4%

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

   7. metadata-eval 78.4%

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

6. Simplified 78.4%

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

   1. add-cbrt-cube 78.4%

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

   2. add-sqr-sqrt 78.4%

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

   3. pow1 78.4%

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

   4. pow1/2 78.4%

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

   5. pow-prod-up 78.4%

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

   6. associate-+r+ 78.4%

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

   7. metadata-eval 78.4%

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

   8. distribute-lft-in 78.4%

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

   9. metadata-eval 78.4%

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

   10. metadata-eval 78.4%

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

8. Applied egg-rr 78.4%

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

9. Taylor expanded in x around inf 37.2%

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

10. Final simplification 37.2%

    \[\leadsto 1 \]
11. Add Preprocessing

Developer target: 79.9% 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 2024112 
(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))))))))