Given's Rotation SVD example

Percentage Accurate: 80.0% → 99.7%

Time: 9.8s

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: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

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 (* (+ 1.0 (/ x (hypot x (* p_m 2.0)))) 0.5))))

C

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

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(Float64(1.0 + Float64(x / hypot(x, Float64(p_m * 2.0)))) * 0.5));
	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(((1.0 + (x / hypot(x, (p_m * 2.0)))) * 0.5));
	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[(N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 15.7%

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

   3. Simplified 15.7%

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

   4. Taylor expanded in x around -inf 57.5%

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

      1. mul-1-neg 57.5%

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

   6. Simplified 57.5%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. fma-udef 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt1-in 99.9%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 88.7%

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

Alternative 2: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -3.4e-24) (/ (- p_m) x) (if (<= x 1.1e-49) (sqrt 0.5) 1.0)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -3.4e-24) {
		tmp = -p_m / x;
	} else if (x <= 1.1e-49) {
		tmp = sqrt(0.5);
	} 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 <= (-3.4d-24)) then
        tmp = -p_m / x
    else if (x <= 1.1d-49) then
        tmp = sqrt(0.5d0)
    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 <= -3.4e-24) {
		tmp = -p_m / x;
	} else if (x <= 1.1e-49) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -3.4e-24:
		tmp = -p_m / x
	elif x <= 1.1e-49:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -3.4e-24)
		tmp = Float64(Float64(-p_m) / x);
	elseif (x <= 1.1e-49)
		tmp = sqrt(0.5);
	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 <= -3.4e-24)
		tmp = -p_m / x;
	elseif (x <= 1.1e-49)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -3.4e-24], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[x, 1.1e-49], N[Sqrt[0.5], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.39999999999999992e-24

   1. Initial program 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in x around -inf 40.5%

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

      1. mul-1-neg 40.5%

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

   6. Simplified 40.5%

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

   if -3.39999999999999992e-24 < x < 1.09999999999999995e-49

   1. Initial program 81.2%

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

   2. Taylor expanded in x around 0 72.4%

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

   if 1.09999999999999995e-49 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 70.2%

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

      1. associate-*r/ 70.2%

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

      2. *-commutative 70.2%

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

   6. Simplified 70.2%

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

      1. unpow2 70.2%

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

      2. times-frac 70.2%

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

   8. Applied egg-rr 70.2%

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

   9. Taylor expanded in p around 0 74.8%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-49}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 55.5% accurate, 35.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -1.4e-145) (/ (- p_m) x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.4e-145) {
		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 <= (-1.4d-145)) 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 <= -1.4e-145) {
		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 <= -1.4e-145:
		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 <= -1.4e-145)
		tmp = Float64(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 <= -1.4e-145)
		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, -1.4e-145], N[((-p$95$m) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.4000000000000001e-145

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in x around -inf 31.2%

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

      1. mul-1-neg 31.2%

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

   6. Simplified 31.2%

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

   if -1.4000000000000001e-145 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 51.9%

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

      1. associate-*r/ 51.9%

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

      2. *-commutative 51.9%

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

   6. Simplified 51.9%

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

      1. unpow2 51.9%

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

      2. times-frac 51.9%

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

   8. Applied egg-rr 51.9%

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

   9. Taylor expanded in p around 0 60.2%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 36.6% 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 77.5%

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

3. Simplified 77.5%

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

4. Taylor expanded in x around inf 26.8%

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

   1. associate-*r/ 26.8%

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

   2. *-commutative 26.8%

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

6. Simplified 26.8%

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

   1. unpow2 26.8%

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

   2. times-frac 26.8%

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

8. Applied egg-rr 26.8%

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

9. Taylor expanded in p around 0 35.6%

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

10. Final simplification 35.6%

    \[\leadsto 1 \]

Developer target: 80.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x))))))

C

double code(double p, double x) {
	return sqrt((0.5 + (copysign(0.5, x) / hypot(1.0, ((2.0 * p) / x)))));
}

Java

public static double code(double p, double x) {
	return Math.sqrt((0.5 + (Math.copySign(0.5, x) / Math.hypot(1.0, ((2.0 * p) / x)))));
}

Python

def code(p, x):
	return math.sqrt((0.5 + (math.copysign(0.5, x) / math.hypot(1.0, ((2.0 * p) / x)))))

Julia

function code(p, x)
	return sqrt(Float64(0.5 + Float64(copysign(0.5, x) / hypot(1.0, Float64(Float64(2.0 * p) / x)))))
end

MATLAB

function tmp = code(p, x)
	tmp = sqrt((0.5 + ((sign(x) * abs(0.5)) / hypot(1.0, ((2.0 * p) / x)))));
end

Wolfram

code[p_, x_] := N[Sqrt[N[(0.5 + N[(N[With[{TMP1 = Abs[0.5], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(2.0 * p), $MachinePrecision] / x), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2023336 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :herbie-target
  (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))))))))