Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.7%

Time: 15.0s

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: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ 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)))) <= -1.0) {
		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)))) <= -1.0) {
		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)))) <= -1.0:
		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)))) <= -1.0)
		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)))) <= -1.0)
		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], -1.0], 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)))) < -1

   1. Initial program 13.8%

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

   3. Taylor expanded in x around -inf 63.4%

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

      1. mul-1-neg 63.4%

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

      2. associate-/l* 63.5%

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

      3. distribute-rgt-neg-in 63.5%

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

      4. associate-/l* 63.8%

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

   5. Simplified 63.8%

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

      1. distribute-rgt-neg-out 63.8%

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

      2. neg-sub0 63.8%

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

      3. associate-*r/ 63.5%

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

      4. sqrt-unprod 64.2%

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

      5. metadata-eval 64.2%

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

      6. metadata-eval 64.2%

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

   7. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{0 - p \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 64.2%

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

      2. associate-*r/ 64.5%

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

      3. *-rgt-identity 64.5%

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

      4. distribute-neg-frac2 64.5%

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

   9. Simplified 64.5%

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

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

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. hypot-define 99.3%

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

      3. associate-*l* 99.3%

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

      4. sqrt-prod 99.3%

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

      5. metadata-eval 99.3%

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

      6. sqrt-unprod 43.9%

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

      7. add-sqr-sqrt 99.4%

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

   4. Applied egg-rr 99.4%

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

4. Final simplification 91.2%

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

Alternative 2: 69.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{p\_m}{-x} - -1.5 \cdot {\left(\frac{p\_m}{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.3e-37)
   1.0
   (if (<= p_m 2.5e-24)
     (- (/ p_m (- x)) (* -1.5 (pow (/ p_m x) 3.0)))
     (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.3e-37) {
		tmp = 1.0;
	} else if (p_m <= 2.5e-24) {
		tmp = (p_m / -x) - (-1.5 * pow((p_m / x), 3.0));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.3d-37) then
        tmp = 1.0d0
    else if (p_m <= 2.5d-24) then
        tmp = (p_m / -x) - ((-1.5d0) * ((p_m / x) ** 3.0d0))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.3e-37) {
		tmp = 1.0;
	} else if (p_m <= 2.5e-24) {
		tmp = (p_m / -x) - (-1.5 * Math.pow((p_m / x), 3.0));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.3e-37:
		tmp = 1.0
	elif p_m <= 2.5e-24:
		tmp = (p_m / -x) - (-1.5 * math.pow((p_m / x), 3.0))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.3e-37)
		tmp = 1.0;
	elseif (p_m <= 2.5e-24)
		tmp = Float64(Float64(p_m / Float64(-x)) - Float64(-1.5 * (Float64(p_m / x) ^ 3.0)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.3e-37)
		tmp = 1.0;
	elseif (p_m <= 2.5e-24)
		tmp = (p_m / -x) - (-1.5 * ((p_m / x) ^ 3.0));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.3e-37], 1.0, If[LessEqual[p$95$m, 2.5e-24], N[(N[(p$95$m / (-x)), $MachinePrecision] - N[(-1.5 * N[Power[N[(p$95$m / x), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.2999999999999999e-37

   1. Initial program 77.9%

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

   3. Taylor expanded in x around inf 46.9%

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

   if 1.2999999999999999e-37 < p < 2.4999999999999999e-24

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{0.5 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{{x}^{2}} + 2 \cdot {p}^{2}}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{2 \cdot {p}^{2} + 0.5 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{{x}^{2}}}}{{x}^{2}}} \]

      2. fma-define 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(2, {p}^{2}, 0.5 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{{x}^{2}}\right)}}{{x}^{2}}} \]

      3. distribute-rgt-out 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, {p}^{2}, 0.5 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{{x}^{2}}\right)}{{x}^{2}}} \]

      4. associate-/l* 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, {p}^{2}, 0.5 \cdot \color{blue}{\left({p}^{4} \cdot \frac{-16 + 4}{{x}^{2}}\right)}\right)}{{x}^{2}}} \]

      5. metadata-eval 100.0%

         \[\leadsto \sqrt{0.5 \cdot \frac{\mathsf{fma}\left(2, {p}^{2}, 0.5 \cdot \left({p}^{4} \cdot \frac{\color{blue}{-12}}{{x}^{2}}\right)\right)}{{x}^{2}}} \]

   5. Simplified 100.0%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{\mathsf{fma}\left(2, {p}^{2}, 0.5 \cdot \left({p}^{4} \cdot \frac{-12}{{x}^{2}}\right)\right)}{{x}^{2}}}} \]

   6. Taylor expanded in x around -inf 98.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{-3 \cdot \frac{{p}^{3} \cdot \sqrt{0.5}}{{x}^{2} \cdot \sqrt{2}} + p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x}} \]
   7. Step-by-step derivation

      1. associate-/l* 98.1%

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

      2. cube-mult 98.1%

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

      3. *-rgt-identity 98.1%

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

      4. metadata-eval 98.1%

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

      5. metadata-eval 98.1%

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

      6. sqrt-unprod 98.1%

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

      7. *-rgt-identity 98.1%

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

      8. metadata-eval 98.1%

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

      9. metadata-eval 98.1%

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

      10. sqrt-unprod 98.1%

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

      11. associate-*l* 98.1%

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

   8. Applied egg-rr 98.1%

      \[\leadsto -1 \cdot \frac{-3 \cdot \color{blue}{\left(p \cdot \left({p}^{2} \cdot \left(\frac{1}{{x}^{2}} \cdot 0.5\right)\right)\right)} + p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]
   9. Step-by-step derivation

      1. associate-*r* 98.1%

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

      2. unpow2 98.1%

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

      3. cube-mult 98.1%

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

      4. associate-*l/ 98.1%

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

      5. metadata-eval 98.1%

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

   10. Simplified 98.1%

       \[\leadsto -1 \cdot \frac{-3 \cdot \color{blue}{\left({p}^{3} \cdot \frac{0.5}{{x}^{2}}\right)} + p \cdot \left(\sqrt{0.5} \cdot \sqrt{2}\right)}{x} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 98.1%

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

       2. *-commutative 98.1%

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

       3. sqrt-unprod 100.0%

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

       4. metadata-eval 100.0%

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

       5. metadata-eval 100.0%

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

       6. *-rgt-identity 100.0%

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

       7. fma-define 100.0%

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

       8. *-commutative 100.0%

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

       9. div-inv 100.0%

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

       10. associate-*l* 100.0%

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

       11. pow-flip 100.0%

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

       12. metadata-eval 100.0%

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

   12. Applied egg-rr 100.0%

       \[\leadsto -1 \cdot \color{blue}{\left(1 \cdot \frac{\mathsf{fma}\left(0.5 \cdot \left({x}^{-2} \cdot {p}^{3}\right), -3, p\right)}{x}\right)} \]
   13. Step-by-step derivation

       1. *-lft-identity 100.0%

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

       2. fma-undefine 100.0%

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

       3. +-commutative 100.0%

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

       4. *-commutative 100.0%

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

       5. associate-*l* 100.0%

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

       6. *-commutative 100.0%

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

       7. metadata-eval 100.0%

          \[\leadsto -1 \cdot \frac{p + \left({p}^{3} \cdot {x}^{-2}\right) \cdot \color{blue}{-1.5}}{x} \]

   14. Simplified 100.0%

       \[\leadsto -1 \cdot \color{blue}{\frac{p + \left({p}^{3} \cdot {x}^{-2}\right) \cdot -1.5}{x}} \]

   15. Taylor expanded in p around 0 98.4%

       \[\leadsto -1 \cdot \color{blue}{\left(p \cdot \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}} + \frac{1}{x}\right)\right)} \]
   16. Step-by-step derivation

       1. +-commutative 98.4%

          \[\leadsto -1 \cdot \left(p \cdot \color{blue}{\left(\frac{1}{x} + -1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right)}\right) \]

       2. distribute-rgt-in 98.4%

          \[\leadsto -1 \cdot \color{blue}{\left(\frac{1}{x} \cdot p + \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot p\right)} \]

       3. associate-*l/ 100.0%

          \[\leadsto -1 \cdot \left(\color{blue}{\frac{1 \cdot p}{x}} + \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot p\right) \]

       4. *-lft-identity 100.0%

          \[\leadsto -1 \cdot \left(\frac{\color{blue}{p}}{x} + \left(-1.5 \cdot \frac{{p}^{2}}{{x}^{3}}\right) \cdot p\right) \]

       5. associate-*r/ 100.0%

          \[\leadsto -1 \cdot \left(\frac{p}{x} + \color{blue}{\frac{-1.5 \cdot {p}^{2}}{{x}^{3}}} \cdot p\right) \]

       6. associate-*l/ 100.0%

          \[\leadsto -1 \cdot \left(\frac{p}{x} + \color{blue}{\frac{\left(-1.5 \cdot {p}^{2}\right) \cdot p}{{x}^{3}}}\right) \]

       7. associate-*r* 100.0%

          \[\leadsto -1 \cdot \left(\frac{p}{x} + \frac{\color{blue}{-1.5 \cdot \left({p}^{2} \cdot p\right)}}{{x}^{3}}\right) \]

       8. unpow2 100.0%

          \[\leadsto -1 \cdot \left(\frac{p}{x} + \frac{-1.5 \cdot \left(\color{blue}{\left(p \cdot p\right)} \cdot p\right)}{{x}^{3}}\right) \]

       9. unpow3 100.0%

          \[\leadsto -1 \cdot \left(\frac{p}{x} + \frac{-1.5 \cdot \color{blue}{{p}^{3}}}{{x}^{3}}\right) \]

       10. associate-*r/ 100.0%

           \[\leadsto -1 \cdot \left(\frac{p}{x} + \color{blue}{-1.5 \cdot \frac{{p}^{3}}{{x}^{3}}}\right) \]

       11. cube-div 100.0%

           \[\leadsto -1 \cdot \left(\frac{p}{x} + -1.5 \cdot \color{blue}{{\left(\frac{p}{x}\right)}^{3}}\right) \]

   17. Simplified 100.0%

       \[\leadsto -1 \cdot \color{blue}{\left(\frac{p}{x} + -1.5 \cdot {\left(\frac{p}{x}\right)}^{3}\right)} \]

   if 2.4999999999999999e-24 < p

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0 83.5%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{p}{-x} - -1.5 \cdot {\left(\frac{p}{x}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.45e-37) 1.0 (if (<= p_m 2.5e-24) (/ p_m (- x)) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.45e-37) {
		tmp = 1.0;
	} else if (p_m <= 2.5e-24) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.45d-37) then
        tmp = 1.0d0
    else if (p_m <= 2.5d-24) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.45e-37) {
		tmp = 1.0;
	} else if (p_m <= 2.5e-24) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.45e-37:
		tmp = 1.0
	elif p_m <= 2.5e-24:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.45e-37)
		tmp = 1.0;
	elseif (p_m <= 2.5e-24)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.45e-37)
		tmp = 1.0;
	elseif (p_m <= 2.5e-24)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.45e-37], 1.0, If[LessEqual[p$95$m, 2.5e-24], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.45000000000000002e-37

   1. Initial program 77.9%

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

   3. Taylor expanded in x around inf 46.9%

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

   if 1.45000000000000002e-37 < p < 2.4999999999999999e-24

   1. Initial program 3.7%

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

   3. Taylor expanded in x around -inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. associate-/l* 97.8%

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

      3. distribute-rgt-neg-in 97.8%

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

      4. associate-/l* 98.4%

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

   5. Simplified 98.4%

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

      1. distribute-rgt-neg-out 98.4%

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

      2. neg-sub0 98.4%

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

      3. associate-*r/ 97.8%

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

      4. sqrt-unprod 98.4%

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

      5. metadata-eval 98.4%

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

      6. metadata-eval 98.4%

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

   7. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{0 - p \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 98.4%

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

      2. associate-*r/ 100.0%

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

      3. *-rgt-identity 100.0%

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

      4. distribute-neg-frac2 100.0%

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

   9. Simplified 100.0%

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

   if 2.4999999999999999e-24 < p

   1. Initial program 86.7%

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

   3. Taylor expanded in x around 0 83.5%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.45 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 2.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.8e-83) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.8e-83) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 2.8d-83) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.8e-83) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.8e-83:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.8e-83)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 2.8e-83)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 2.8e-83], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 2.8000000000000001e-83

   1. Initial program 78.8%

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

   3. Taylor expanded in x around -inf 15.6%

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

      1. mul-1-neg 15.6%

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

      2. associate-/l* 15.6%

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

      3. distribute-rgt-neg-in 15.6%

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

      4. associate-/l* 15.7%

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

   5. Simplified 15.7%

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

      1. distribute-rgt-neg-out 15.7%

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

      2. neg-sub0 15.7%

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

      3. associate-*r/ 15.6%

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

      4. sqrt-unprod 15.8%

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

      5. metadata-eval 15.8%

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

      6. metadata-eval 15.8%

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

   7. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{0 - p \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 15.8%

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

      2. associate-*r/ 15.8%

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

      3. *-rgt-identity 15.8%

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

      4. distribute-neg-frac2 15.8%

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

   9. Simplified 15.8%

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

   if 2.8000000000000001e-83 < p

   1. Initial program 80.5%

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

   3. Taylor expanded in x around 0 74.6%

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

Alternative 5: 28.4% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -1.2e-149) (/ p_m (- x)) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -1.2e-149) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.2d-149)) then
        tmp = p_m / -x
    else
        tmp = p_m / x
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -1.2e-149) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -1.2e-149:
		tmp = p_m / -x
	else:
		tmp = p_m / x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -1.2e-149)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(p_m / x);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -1.2e-149)
		tmp = p_m / -x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -1.2e-149], N[(p$95$m / (-x)), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2000000000000001e-149

   1. Initial program 57.2%

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

   3. Taylor expanded in x around -inf 32.0%

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

      1. mul-1-neg 32.0%

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

      2. associate-/l* 32.1%

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

      3. distribute-rgt-neg-in 32.1%

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

      4. associate-/l* 32.2%

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

   5. Simplified 32.2%

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

      1. distribute-rgt-neg-out 32.2%

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

      2. neg-sub0 32.2%

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

      3. associate-*r/ 32.1%

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

      4. sqrt-unprod 32.4%

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

      5. metadata-eval 32.4%

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

      6. metadata-eval 32.4%

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

   7. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{0 - p \cdot \frac{1}{x}} \]
   8. Step-by-step derivation

      1. neg-sub0 32.4%

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

      2. associate-*r/ 32.5%

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

      3. *-rgt-identity 32.5%

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

      4. distribute-neg-frac2 32.5%

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

   9. Simplified 32.5%

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

   if -1.2000000000000001e-149 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around -inf 3.9%

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

      1. mul-1-neg 3.9%

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

      2. associate-/l* 3.9%

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

      3. distribute-rgt-neg-in 3.9%

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

      4. associate-/l* 3.9%

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

   5. Simplified 3.9%

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

      1. pow1 3.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 3.0%

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

      4. sqr-neg 3.0%

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

      5. sqrt-unprod 3.0%

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

      6. add-sqr-sqrt 3.1%

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

      7. associate-*r/ 3.1%

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

      8. sqrt-unprod 3.1%

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

      9. metadata-eval 3.1%

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

      10. metadata-eval 3.1%

          \[\leadsto {\left(p \cdot \frac{\color{blue}{1}}{x}\right)}^{1} \]

   7. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{{\left(p \cdot \frac{1}{x}\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 3.1%

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

      2. associate-*r/ 3.1%

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

      3. *-rgt-identity 3.1%

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

   9. Simplified 3.1%

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

Alternative 6: 6.2% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in x around -inf 17.5%

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

   1. mul-1-neg 17.5%

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

   2. associate-/l* 17.5%

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

   3. distribute-rgt-neg-in 17.5%

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

   4. associate-/l* 17.6%

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

5. Simplified 17.6%

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

   1. pow1 17.6%

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

   2. add-sqr-sqrt 15.6%

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

   3. sqrt-unprod 17.2%

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

   4. sqr-neg 17.2%

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

   5. sqrt-unprod 1.6%

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

   6. add-sqr-sqrt 14.3%

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

   7. associate-*r/ 14.2%

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

   8. sqrt-unprod 14.3%

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

   9. metadata-eval 14.3%

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

   10. metadata-eval 14.3%

       \[\leadsto {\left(p \cdot \frac{\color{blue}{1}}{x}\right)}^{1} \]

7. Applied egg-rr 14.3%

   \[\leadsto \color{blue}{{\left(p \cdot \frac{1}{x}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 14.3%

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

   2. associate-*r/ 14.4%

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

   3. *-rgt-identity 14.4%

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

9. Simplified 14.4%

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

Developer target: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024107 
(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))))))))