Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.9%

Time: 9.2s

Alternatives: 5

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: 79.6% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9999)
   (/ (- p) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	}
	return tmp;
}

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((p * 2.0), x)))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.9999)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(Float64(p * 2.0), x)))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((p * 2.0), x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p * N[(4.0 * p), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.9999], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[N[(p * 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)))) < -0.99990000000000001

   1. Initial program 15.4%

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

      1. add-sqr-sqrt 15.4%

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

      2. hypot-def 15.4%

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

      3. associate-*l* 15.4%

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

      4. sqrt-prod 15.4%

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

      5. metadata-eval 15.4%

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

      6. sqrt-unprod 10.8%

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

      7. add-sqr-sqrt 15.4%

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

   3. Applied egg-rr 15.4%

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

      1. add-cbrt-cube 15.4%

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

      2. pow1/3 15.4%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 15.4%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 15.4%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 15.4%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 15.4%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 15.4%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 15.4%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 15.4%

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

   5. Applied egg-rr 15.4%

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

      1. unpow1/3 15.4%

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

      2. *-commutative 15.4%

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

   7. Simplified 15.4%

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

   8. Taylor expanded in x around -inf 62.6%

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

      1. mul-1-neg 62.6%

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

      2. distribute-neg-frac 62.6%

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

   10. Simplified 62.6%

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

   if -0.99990000000000001 < (/.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. 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 42.6%

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

      7. add-sqr-sqrt 99.9%

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

   3. 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 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 2: 68.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 1.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= p 1.6e-41) (/ (- p) x) (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 1.6e-41) {
		tmp = -p / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 1.6d-41) then
        tmp = -p / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 1.6e-41) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 1.6e-41:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 1.6e-41)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 1.6e-41)
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 1.6e-41], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.60000000000000006e-41

   1. Initial program 79.8%

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

      1. add-sqr-sqrt 79.8%

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

      2. hypot-def 79.8%

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

      3. associate-*l* 79.8%

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

      4. sqrt-prod 79.8%

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

      5. metadata-eval 79.8%

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

      6. sqrt-unprod 17.7%

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

      7. add-sqr-sqrt 79.8%

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

   3. Applied egg-rr 79.8%

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

      1. add-cbrt-cube 79.8%

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

      2. pow1/3 79.8%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 79.8%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 79.8%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 79.8%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 79.8%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 79.8%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 79.8%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 79.8%

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

   5. Applied egg-rr 79.8%

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

      1. unpow1/3 79.8%

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

      2. *-commutative 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in x around -inf 16.7%

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

      1. mul-1-neg 16.7%

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

      2. distribute-neg-frac 16.7%

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

   10. Simplified 16.7%

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

   if 1.60000000000000006e-41 < p

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 79.4%

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

4. Final simplification 32.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 3: 69.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;p \leq 1.95 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= p 1.95e-31) 1.0 (sqrt 0.5)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (p <= 1.95e-31) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= 1.95d-31) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (p <= 1.95e-31) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if p <= 1.95e-31:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (p <= 1.95e-31)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= 1.95e-31)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[p, 1.95e-31], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 1.9500000000000001e-31

   1. Initial program 79.9%

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

   2. Taylor expanded in x around inf 45.8%

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

   if 1.9500000000000001e-31 < p

   1. Initial program 91.4%

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

   2. Taylor expanded in x around 0 80.3%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.95 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 4: 28.6% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -5e-310) (/ (- p) x) (/ p x)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p / x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = -p / x
    else
        tmp = p / x
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = -p / x;
	} else {
		tmp = p / x;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -5e-310:
		tmp = -p / x
	else:
		tmp = p / x
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = Float64(p / x);
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = -p / x;
	else
		tmp = p / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -5e-310], N[((-p) / x), $MachinePrecision], N[(p / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 64.2%

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

      1. add-sqr-sqrt 64.2%

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

      2. hypot-def 64.2%

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

      3. associate-*l* 64.2%

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

      4. sqrt-prod 64.2%

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

      5. metadata-eval 64.2%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 64.2%

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

   3. Applied egg-rr 64.2%

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

      1. add-cbrt-cube 64.1%

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

      2. pow1/3 64.2%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 64.2%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 64.2%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 64.2%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 64.2%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 64.2%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 64.2%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 64.2%

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

   5. Applied egg-rr 64.2%

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

      1. unpow1/3 64.1%

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

      2. *-commutative 64.1%

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

   7. Simplified 64.1%

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

   8. Taylor expanded in x around -inf 28.2%

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

      1. mul-1-neg 28.2%

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

      2. distribute-neg-frac 28.2%

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

   10. Simplified 28.2%

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

   if -4.999999999999985e-310 < 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. hypot-def 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 45.1%

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

      7. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow1/3 100.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 100.0%

         \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      4. pow1 100.0%

         \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

      5. pow1/2 100.0%

         \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 100.0%

         \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. distribute-lft-in 100.0%

         \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      8. metadata-eval 100.0%

         \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

      9. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow1/3 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around -inf 4.4%

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

      1. unpow2 4.4%

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

      2. associate-*r/ 4.6%

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

      3. unpow2 4.6%

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

      4. associate-/r* 4.6%

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

   10. Simplified 4.6%

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

   11. Taylor expanded in p around 0 3.3%

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

4. Final simplification 15.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]

Alternative 5: 6.7% accurate, 71.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \frac{p}{x} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (/ p x))

C

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

Fortran

NOTE: p should be positive before calling this function
real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    code = p / x
end function

Java

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

Python

p = abs(p)
def code(p, x):
	return p / x

Julia

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

MATLAB

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

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. add-sqr-sqrt 82.8%

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

   2. hypot-def 82.8%

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

   3. associate-*l* 82.8%

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

   4. sqrt-prod 82.8%

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

   5. metadata-eval 82.8%

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

   6. sqrt-unprod 36.2%

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

   7. add-sqr-sqrt 82.8%

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

3. Applied egg-rr 82.8%

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

   1. add-cbrt-cube 82.8%

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

   2. pow1/3 82.8%

      \[\leadsto \color{blue}{{\left(\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 82.8%

      \[\leadsto {\left(\color{blue}{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

   4. pow1 82.8%

      \[\leadsto {\left(\color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1}} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}\right)}^{0.3333333333333333} \]

   5. pow1/2 82.8%

      \[\leadsto {\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{1} \cdot \color{blue}{{\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

   6. pow-prod-up 82.8%

      \[\leadsto {\color{blue}{\left({\left(0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

   7. distribute-lft-in 82.8%

      \[\leadsto {\left({\color{blue}{\left(0.5 \cdot 1 + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   8. metadata-eval 82.8%

      \[\leadsto {\left({\left(\color{blue}{0.5} + 0.5 \cdot \frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

   9. metadata-eval 82.8%

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

5. Applied egg-rr 82.8%

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

   1. unpow1/3 82.7%

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

   2. *-commutative 82.7%

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

7. Simplified 82.7%

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

8. Taylor expanded in x around -inf 12.4%

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

   1. unpow2 12.4%

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

   2. associate-*r/ 14.0%

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

   3. unpow2 14.0%

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

   4. associate-/r* 14.0%

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

10. Simplified 14.0%

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

11. Taylor expanded in p around 0 12.8%

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

12. Final simplification 12.8%

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

Developer target: 79.6% 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 2023240 
(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))))))))