Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.8%

Time: 10.0s

Alternatives: 7

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: 79.3% 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, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -0.99996:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p_m \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.99996)
   (/ (- 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)))) <= -0.99996) {
		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)))) <= -0.99996) {
		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)))) <= -0.99996:
		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)))) <= -0.99996)
		tmp = Float64(Float64(-p_m) / 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)))) <= -0.99996)
		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], -0.99996], 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 4 p) p) (*.f64 x x)))) < -0.99995999999999996

   1. Initial program 18.6%

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

   3. Taylor expanded in x around -inf 59.5%

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

   4. Taylor expanded in p around -inf 70.4%

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

      1. associate-*r/ 70.4%

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

      2. mul-1-neg 70.4%

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

   6. Simplified 70.4%

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

   if -0.99995999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.9%

      \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
   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-def 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 47.4%

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

      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 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.99996:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;p_m \cdot \sqrt{{x}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.2e-62) (* p_m (sqrt (pow x -2.0))) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.2e-62) {
		tmp = p_m * sqrt(pow(x, -2.0));
	} 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 <= 2.2d-62) then
        tmp = p_m * sqrt((x ** (-2.0d0)))
    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 <= 2.2e-62) {
		tmp = p_m * Math.sqrt(Math.pow(x, -2.0));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 2.2e-62:
		tmp = p_m * math.sqrt(math.pow(x, -2.0))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 2.2e-62)
		tmp = Float64(p_m * sqrt((x ^ -2.0)));
	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 <= 2.2e-62)
		tmp = p_m * sqrt((x ^ -2.0));
	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, 2.2e-62], N[(p$95$m * N[Sqrt[N[Power[x, -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 2.20000000000000017e-62

   1. Initial program 74.8%

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

   3. Taylor expanded in x around -inf 20.8%

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

      1. pow1/2 20.8%

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

      2. associate-*r* 20.8%

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

      3. metadata-eval 20.8%

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

      4. *-un-lft-identity 20.8%

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

      5. div-inv 20.8%

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

      6. unpow-prod-down 24.8%

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

      7. pow1/2 24.8%

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

      8. unpow2 24.8%

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

      9. sqrt-prod 19.7%

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

      10. add-sqr-sqrt 22.4%

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

      11. pow-flip 22.4%

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

      12. metadata-eval 22.4%

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

   5. Applied egg-rr 22.4%

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

      1. unpow1/2 22.4%

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

   7. Simplified 22.4%

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

   if 2.20000000000000017e-62 < p

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0 80.2%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.2 \cdot 10^{-62}:\\ \;\;\;\;p \cdot \sqrt{{x}^{-2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p_m \leq 2.5 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 4.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{elif}\;p_m \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 3.8 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\frac{p_m \cdot p_m}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 2.5e-205)
   1.0
   (if (<= p_m 4.1e-102)
     (/ (- p_m) x)
     (if (<= p_m 6.5e-77)
       1.0
       (if (<= p_m 3.8e-66) (sqrt (/ (* p_m p_m) (* x x))) (sqrt 0.5))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 2.5e-205) {
		tmp = 1.0;
	} else if (p_m <= 4.1e-102) {
		tmp = -p_m / x;
	} else if (p_m <= 6.5e-77) {
		tmp = 1.0;
	} else if (p_m <= 3.8e-66) {
		tmp = sqrt(((p_m * p_m) / (x * 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 <= 2.5d-205) then
        tmp = 1.0d0
    else if (p_m <= 4.1d-102) then
        tmp = -p_m / x
    else if (p_m <= 6.5d-77) then
        tmp = 1.0d0
    else if (p_m <= 3.8d-66) then
        tmp = sqrt(((p_m * p_m) / (x * 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 <= 2.5e-205) {
		tmp = 1.0;
	} else if (p_m <= 4.1e-102) {
		tmp = -p_m / x;
	} else if (p_m <= 6.5e-77) {
		tmp = 1.0;
	} else if (p_m <= 3.8e-66) {
		tmp = Math.sqrt(((p_m * p_m) / (x * 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 <= 2.5e-205:
		tmp = 1.0
	elif p_m <= 4.1e-102:
		tmp = -p_m / x
	elif p_m <= 6.5e-77:
		tmp = 1.0
	elif p_m <= 3.8e-66:
		tmp = math.sqrt(((p_m * p_m) / (x * 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 <= 2.5e-205)
		tmp = 1.0;
	elseif (p_m <= 4.1e-102)
		tmp = Float64(Float64(-p_m) / x);
	elseif (p_m <= 6.5e-77)
		tmp = 1.0;
	elseif (p_m <= 3.8e-66)
		tmp = sqrt(Float64(Float64(p_m * p_m) / Float64(x * 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 <= 2.5e-205)
		tmp = 1.0;
	elseif (p_m <= 4.1e-102)
		tmp = -p_m / x;
	elseif (p_m <= 6.5e-77)
		tmp = 1.0;
	elseif (p_m <= 3.8e-66)
		tmp = sqrt(((p_m * p_m) / (x * 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, 2.5e-205], 1.0, If[LessEqual[p$95$m, 4.1e-102], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[p$95$m, 6.5e-77], 1.0, If[LessEqual[p$95$m, 3.8e-66], N[Sqrt[N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 2.5e-205 or 4.1000000000000003e-102 < p < 6.4999999999999999e-77

   1. Initial program 83.3%

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

      1. add-cbrt-cube 83.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      2. pow1/3 83.3%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. fma-udef 83.3%

         \[\leadsto {\left({\color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

      2. associate-*l/ 83.3%

         \[\leadsto {\left({\left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0.5\right)}^{1.5}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 83.3%

      \[\leadsto {\left({\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 39.9%

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

   if 2.5e-205 < p < 4.1000000000000003e-102

   1. Initial program 33.8%

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

   3. Taylor expanded in x around -inf 35.8%

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

   4. Taylor expanded in p around -inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. mul-1-neg 72.3%

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

   6. Simplified 72.3%

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

   if 6.4999999999999999e-77 < p < 3.7999999999999998e-66

   1. Initial program 4.1%

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

   3. Taylor expanded in x around -inf 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. unpow2 99.6%

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

      3. times-frac 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. associate-*l/ 99.2%

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

      2. *-lft-identity 99.2%

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

   7. Simplified 99.2%

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

      1. associate-*r* 99.2%

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

      2. metadata-eval 99.2%

         \[\leadsto \sqrt{\color{blue}{1} \cdot \frac{\frac{{p}^{2}}{x}}{x}} \]

      3. *-un-lft-identity 99.2%

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

      4. associate-/l/ 99.6%

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

      5. unpow2 99.6%

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

      6. frac-times 100.0%

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

      7. frac-2neg 100.0%

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

      8. frac-2neg 100.0%

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

      9. frac-times 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 3.7999999999999998e-66 < p

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0 79.5%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 2.5 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 6.5 \cdot 10^{-77}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 3.8 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{\frac{p \cdot p}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 4.6 \cdot 10^{-207}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 9 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 2.5 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 4.6e-207)
     1.0
     (if (<= p_m 9e-106)
       t_0
       (if (<= p_m 1.3e-70) 1.0 (if (<= p_m 2.5e-62) t_0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 4.6e-207) {
		tmp = 1.0;
	} else if (p_m <= 9e-106) {
		tmp = t_0;
	} else if (p_m <= 1.3e-70) {
		tmp = 1.0;
	} else if (p_m <= 2.5e-62) {
		tmp = t_0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 4.6d-207) then
        tmp = 1.0d0
    else if (p_m <= 9d-106) then
        tmp = t_0
    else if (p_m <= 1.3d-70) then
        tmp = 1.0d0
    else if (p_m <= 2.5d-62) then
        tmp = t_0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 4.6e-207) {
		tmp = 1.0;
	} else if (p_m <= 9e-106) {
		tmp = t_0;
	} else if (p_m <= 1.3e-70) {
		tmp = 1.0;
	} else if (p_m <= 2.5e-62) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 4.6e-207:
		tmp = 1.0
	elif p_m <= 9e-106:
		tmp = t_0
	elif p_m <= 1.3e-70:
		tmp = 1.0
	elif p_m <= 2.5e-62:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 4.6e-207)
		tmp = 1.0;
	elseif (p_m <= 9e-106)
		tmp = t_0;
	elseif (p_m <= 1.3e-70)
		tmp = 1.0;
	elseif (p_m <= 2.5e-62)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 4.6e-207)
		tmp = 1.0;
	elseif (p_m <= 9e-106)
		tmp = t_0;
	elseif (p_m <= 1.3e-70)
		tmp = 1.0;
	elseif (p_m <= 2.5e-62)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 4.6e-207], 1.0, If[LessEqual[p$95$m, 9e-106], t$95$0, If[LessEqual[p$95$m, 1.3e-70], 1.0, If[LessEqual[p$95$m, 2.5e-62], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 4.6000000000000001e-207 or 8.99999999999999911e-106 < p < 1.30000000000000001e-70

   1. Initial program 82.8%

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

      1. add-cbrt-cube 82.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      2. pow1/3 82.8%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 82.8%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. fma-udef 82.8%

         \[\leadsto {\left({\color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

      2. associate-*l/ 82.8%

         \[\leadsto {\left({\left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0.5\right)}^{1.5}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 82.8%

      \[\leadsto {\left({\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 39.7%

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

   if 4.6000000000000001e-207 < p < 8.99999999999999911e-106 or 1.30000000000000001e-70 < p < 2.5000000000000001e-62

   1. Initial program 32.0%

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

   3. Taylor expanded in x around -inf 40.5%

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

   4. Taylor expanded in p around -inf 73.9%

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

      1. associate-*r/ 73.9%

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

      2. mul-1-neg 73.9%

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

   6. Simplified 73.9%

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

   if 2.5000000000000001e-62 < p

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0 80.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.6 \cdot 10^{-207}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 9 \cdot 10^{-106}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.4% accurate, 35.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-187}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -7.6e-187) (/ (- p_m) x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -7.6e-187) {
		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 <= (-7.6d-187)) 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 <= -7.6e-187) {
		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 <= -7.6e-187:
		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 <= -7.6e-187)
		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 <= -7.6e-187)
		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, -7.6e-187], N[((-p$95$m) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.60000000000000051e-187

   1. Initial program 58.8%

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

   3. Taylor expanded in x around -inf 33.0%

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

   4. Taylor expanded in p around -inf 37.1%

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

      1. associate-*r/ 37.1%

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

      2. mul-1-neg 37.1%

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

   6. Simplified 37.1%

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

   if -7.60000000000000051e-187 < 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-cbrt-cube 100.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. fma-udef 100.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

      2. associate-*l/ 100.0%

         \[\leadsto {\left({\left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0.5\right)}^{1.5}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 100.0%

      \[\leadsto {\left({\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 56.0%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-187}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.0% accurate, 42.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+54}:\\ \;\;\;\;\frac{p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 (if (<= x -4e+54) (/ p_m x) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -4e+54) {
		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 <= (-4d+54)) 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 <= -4e+54) {
		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 <= -4e+54:
		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 <= -4e+54)
		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 <= -4e+54)
		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, -4e+54], N[(p$95$m / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000003e54

   1. Initial program 45.7%

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

   3. Taylor expanded in x around -inf 49.8%

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

   4. Taylor expanded in p around 0 45.0%

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

   if -4.0000000000000003e54 < x

   1. Initial program 86.4%

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

      1. add-cbrt-cube 86.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      2. pow1/3 86.4%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

   4. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. fma-udef 86.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

      2. associate-*l/ 86.4%

         \[\leadsto {\left({\left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0.5\right)}^{1.5}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 86.4%

      \[\leadsto {\left({\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

   7. Taylor expanded in x around inf 40.0%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+54}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.7% 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 80.2%

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

   1. add-cbrt-cube 80.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

   2. pow1/3 80.2%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 80.2%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. fma-udef 80.2%

      \[\leadsto {\left({\color{blue}{\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)} \cdot 0.5 + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

   2. associate-*l/ 80.2%

      \[\leadsto {\left({\left(\color{blue}{\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)}} + 0.5\right)}^{1.5}\right)}^{0.3333333333333333} \]

6. Applied egg-rr 80.2%

   \[\leadsto {\left({\color{blue}{\left(\frac{x \cdot 0.5}{\mathsf{hypot}\left(x, 2 \cdot p\right)} + 0.5\right)}}^{1.5}\right)}^{0.3333333333333333} \]

7. Taylor expanded in x around inf 35.1%

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

8. Final simplification 35.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Developer target: 79.3% 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 2024011 
(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))))))))