Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.8%

Time: 13.6s

Alternatives: 6

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: 79.7% 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

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999)
   (/ p_m (- x))
   (sqrt (fma x (/ 0.5 (hypot (* 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)))) <= -0.999) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(fma(x, (0.5 / hypot((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)))) <= -0.999)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(fma(x, Float64(0.5 / hypot(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], -0.999], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(x * N[(0.5 / N[Sqrt[N[(p$95$m * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.998999999999999999

   1. Initial program 19.1%

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

   3. Taylor expanded in x around -inf 50.9%

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

   4. Taylor expanded in p around -inf 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-neg-frac2 57.5%

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

   6. Simplified 57.5%

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

   if -0.998999999999999999 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. hypot-define 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. metadata-eval 99.9%

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

      6. sqrt-unprod 54.0%

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

      7. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*l/ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. fma-define 99.9%

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

      5. *-commutative 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \frac{0.5}{\mathsf{hypot}\left(p \cdot 2, x\right)}, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p\_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999)
   (/ 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)))) <= -0.999) {
		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)))) <= -0.999) {
		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)))) <= -0.999:
		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)))) <= -0.999)
		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)))) <= -0.999)
		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], -0.999], 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)))) < -0.998999999999999999

   1. Initial program 19.1%

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

   3. Taylor expanded in x around -inf 50.9%

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

   4. Taylor expanded in p around -inf 57.5%

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

      1. mul-1-neg 57.5%

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

      2. distribute-neg-frac2 57.5%

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

   6. Simplified 57.5%

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

   if -0.998999999999999999 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. hypot-define 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. metadata-eval 99.9%

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

      6. sqrt-unprod 54.0%

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

      7. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 8.2 \cdot 10^{-72}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= p_m 8.2e-72) 1.0 (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 8.2e-72) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 8.2d-72) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 8.2e-72) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 8.2e-72:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 8.2e-72)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 8.2e-72)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 8.2e-72], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 8.20000000000000007e-72

   1. Initial program 77.7%

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

      1. add-sqr-sqrt 77.7%

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

      2. hypot-define 77.7%

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

      3. associate-*l* 77.7%

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

      4. sqrt-prod 77.7%

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

      5. metadata-eval 77.7%

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

      6. sqrt-unprod 22.7%

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

      7. add-sqr-sqrt 77.7%

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

   4. Applied egg-rr 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. distribute-rgt-in 77.7%

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

      3. metadata-eval 77.7%

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

      4. associate-*l/ 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. *-lft-identity 77.7%

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

      2. +-commutative 77.7%

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

      3. associate-/l* 76.3%

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

      4. fma-define 72.6%

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

      5. *-commutative 72.6%

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

   8. Simplified 72.6%

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

      1. add-log-exp 72.5%

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

   10. Applied egg-rr 72.5%

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

   11. Taylor expanded in x around inf 41.5%

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

   if 8.20000000000000007e-72 < p

   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. Taylor expanded in x around 0 83.6%

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

Alternative 4: 55.3% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -2.7e-146) (/ p_m (- x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.7e-146) {
		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 <= (-2.7d-146)) 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 <= -2.7e-146) {
		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 <= -2.7e-146:
		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 <= -2.7e-146)
		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 <= -2.7e-146)
		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, -2.7e-146], N[(p$95$m / (-x)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.69999999999999995e-146

   1. Initial program 64.5%

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

   3. Taylor expanded in x around -inf 25.7%

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

   4. Taylor expanded in p around -inf 27.1%

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

      1. mul-1-neg 27.1%

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

      2. distribute-neg-frac2 27.1%

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

   6. Simplified 27.1%

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

   if -2.69999999999999995e-146 < 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 57.0%

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

      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%

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

      1. *-un-lft-identity 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*l/ 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. *-lft-identity 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. fma-define 100.0%

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

      5. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around inf 59.7%

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

Alternative 5: 36.6% accurate, 26.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -3.8e+124) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.8e+124) {
		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 <= (-3.8d+124)) 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 <= -3.8e+124) {
		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 <= -3.8e+124:
		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 <= -3.8e+124)
		tmp = Float64(p_m / 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 <= -3.8e+124)
		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, -3.8e+124], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.7999999999999998e124

   1. Initial program 70.7%

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

   3. Taylor expanded in x around -inf 53.9%

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

   4. Taylor expanded in p around 0 53.3%

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

   if -3.7999999999999998e124 < x

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 83.1%

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

      2. hypot-define 83.1%

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

      3. associate-*l* 83.1%

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

      4. sqrt-prod 83.1%

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

      5. metadata-eval 83.1%

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

      6. sqrt-unprod 45.1%

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

      7. add-sqr-sqrt 83.1%

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

   4. Applied egg-rr 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. distribute-rgt-in 83.1%

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

      3. metadata-eval 83.1%

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

      4. associate-*l/ 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. *-lft-identity 83.1%

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

      2. +-commutative 83.1%

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

      3. associate-/l* 82.7%

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

      4. fma-define 81.8%

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

      5. *-commutative 81.8%

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

   8. Simplified 81.8%

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

      1. add-log-exp 81.8%

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

   10. Applied egg-rr 81.8%

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

   11. Taylor expanded in x around inf 38.5%

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

Alternative 6: 36.0% 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 82.2%

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

   1. add-sqr-sqrt 82.2%

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

   2. hypot-define 82.2%

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

   3. associate-*l* 82.2%

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

   4. sqrt-prod 82.2%

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

   5. metadata-eval 82.2%

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

   6. sqrt-unprod 44.8%

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

   7. add-sqr-sqrt 82.2%

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

4. Applied egg-rr 82.2%

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

   1. *-un-lft-identity 82.2%

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

   2. distribute-rgt-in 82.2%

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

   3. metadata-eval 82.2%

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

   4. associate-*l/ 82.2%

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

6. Applied egg-rr 82.2%

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

   1. *-lft-identity 82.2%

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

   2. +-commutative 82.2%

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

   3. associate-/l* 81.2%

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

   4. fma-define 78.8%

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

   5. *-commutative 78.8%

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

8. Simplified 78.8%

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

   1. add-log-exp 78.8%

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

10. Applied egg-rr 78.8%

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

11. Taylor expanded in x around inf 36.6%

    \[\leadsto \color{blue}{1} \]
12. Add Preprocessing

Developer target: 79.7% 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 2024087 
(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))))))))