Given's Rotation SVD example

Percentage Accurate: 79.4% → 99.9%

Time: 9.8s

Alternatives: 6

Speedup: 1.8×

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;{\left(\frac{p}{x}\right)}^{3} \cdot 1.5 - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.98)
   (- (* (pow (/ p x) 3.0) 1.5) (/ p x))
   (sqrt (* 0.5 (log (exp (+ 1.0 (/ x (hypot x (* p 2.0))))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = (pow((p / x), 3.0) * 1.5) - (p / x);
	} else {
		tmp = sqrt((0.5 * log(exp((1.0 + (x / hypot(x, (p * 2.0))))))));
	}
	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)))) <= -0.98) {
		tmp = (Math.pow((p / x), 3.0) * 1.5) - (p / x);
	} else {
		tmp = Math.sqrt((0.5 * Math.log(Math.exp((1.0 + (x / Math.hypot(x, (p * 2.0))))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98:
		tmp = (math.pow((p / x), 3.0) * 1.5) - (p / x)
	else:
		tmp = math.sqrt((0.5 * math.log(math.exp((1.0 + (x / math.hypot(x, (p * 2.0))))))))
	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)))) <= -0.98)
		tmp = Float64(Float64((Float64(p / x) ^ 3.0) * 1.5) - Float64(p / x));
	else
		tmp = sqrt(Float64(0.5 * log(exp(Float64(1.0 + Float64(x / hypot(x, Float64(p * 2.0))))))));
	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)))) <= -0.98)
		tmp = (((p / x) ^ 3.0) * 1.5) - (p / x);
	else
		tmp = sqrt((0.5 * log(exp((1.0 + (x / hypot(x, (p * 2.0))))))));
	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], -0.98], N[(N[(N[Power[N[(p / x), $MachinePrecision], 3.0], $MachinePrecision] * 1.5), $MachinePrecision] - N[(p / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $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)))) < -0.97999999999999998

   1. Initial program 16.8%

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

      1. add-log-exp 16.8%

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

      2. +-commutative 16.8%

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

      3. add-sqr-sqrt 16.8%

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

      4. hypot-def 16.8%

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

      5. associate-*r* 16.8%

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

      6. *-commutative 16.8%

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

      7. sqrt-prod 16.8%

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

      8. sqrt-prod 4.2%

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

      9. add-sqr-sqrt 16.8%

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

      10. metadata-eval 16.8%

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

   3. Applied egg-rr 16.8%

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

      1. add-log-exp 16.8%

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

      2. *-commutative 16.8%

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

      3. add-log-exp 16.8%

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

      4. *-commutative 16.8%

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

      5. distribute-lft-in 16.8%

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

      6. metadata-eval 16.8%

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

   5. Applied egg-rr 16.8%

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

   6. Taylor expanded in x around -inf 44.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + -0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative 44.8%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} + -1 \cdot \frac{p}{x}} \]

      2. fma-def 44.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}}, -1 \cdot \frac{p}{x}\right)} \]

      3. distribute-rgt-out 44.8%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{{p}^{4} \cdot \left(-4 + -2\right)}}{p \cdot {x}^{3}}, -1 \cdot \frac{p}{x}\right) \]

      4. times-frac 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{p}^{4}}{p} \cdot \frac{-4 + -2}{{x}^{3}}}, -1 \cdot \frac{p}{x}\right) \]

      5. metadata-eval 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{\color{blue}{-6}}{{x}^{3}}, -1 \cdot \frac{p}{x}\right) \]

      6. associate-*r/ 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{-6}{{x}^{3}}, \color{blue}{\frac{-1 \cdot p}{x}}\right) \]

      7. mul-1-neg 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{-6}{{x}^{3}}, \frac{\color{blue}{-p}}{x}\right) \]

   8. Simplified 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{-6}{{x}^{3}}, \frac{-p}{x}\right)} \]

   9. Taylor expanded in p around 0 51.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + 1.5 \cdot \frac{{p}^{3}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. +-commutative 51.2%

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

       2. mul-1-neg 51.2%

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

       3. unsub-neg 51.2%

          \[\leadsto \color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{3}} - \frac{p}{x}} \]

       4. *-commutative 51.2%

          \[\leadsto \color{blue}{\frac{{p}^{3}}{{x}^{3}} \cdot 1.5} - \frac{p}{x} \]

       5. cube-div 54.3%

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

   11. Simplified 54.3%

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

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x 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-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. hypot-def 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. sqrt-prod 100.0%

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

      8. sqrt-prod 48.9%

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

      9. add-sqr-sqrt 100.0%

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

      10. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;{\left(\frac{p}{x}\right)}^{3} \cdot 1.5 - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\right)}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;{\left(\frac{p}{x}\right)}^{3} \cdot 1.5 - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.98)
   (- (* (pow (/ p x) 3.0) 1.5) (/ p x))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot x (* p 2.0))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98) {
		tmp = (pow((p / x), 3.0) * 1.5) - (p / x);
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot(x, (p * 2.0))))));
	}
	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)))) <= -0.98) {
		tmp = (Math.pow((p / x), 3.0) * 1.5) - (p / x);
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot(x, (p * 2.0))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.98:
		tmp = (math.pow((p / x), 3.0) * 1.5) - (p / x)
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot(x, (p * 2.0))))))
	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)))) <= -0.98)
		tmp = Float64(Float64((Float64(p / x) ^ 3.0) * 1.5) - Float64(p / x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / hypot(x, Float64(p * 2.0))))));
	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)))) <= -0.98)
		tmp = (((p / x) ^ 3.0) * 1.5) - (p / x);
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot(x, (p * 2.0))))));
	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], -0.98], N[(N[(N[Power[N[(p / x), $MachinePrecision], 3.0], $MachinePrecision] * 1.5), $MachinePrecision] - N[(p / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 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)))) < -0.97999999999999998

   1. Initial program 16.8%

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

      1. add-log-exp 16.8%

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

      2. +-commutative 16.8%

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

      3. add-sqr-sqrt 16.8%

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

      4. hypot-def 16.8%

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

      5. associate-*r* 16.8%

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

      6. *-commutative 16.8%

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

      7. sqrt-prod 16.8%

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

      8. sqrt-prod 4.2%

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

      9. add-sqr-sqrt 16.8%

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

      10. metadata-eval 16.8%

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

   3. Applied egg-rr 16.8%

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

      1. add-log-exp 16.8%

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

      2. *-commutative 16.8%

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

      3. add-log-exp 16.8%

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

      4. *-commutative 16.8%

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

      5. distribute-lft-in 16.8%

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

      6. metadata-eval 16.8%

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

   5. Applied egg-rr 16.8%

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

   6. Taylor expanded in x around -inf 44.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + -0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}}} \]
   7. Step-by-step derivation

      1. +-commutative 44.8%

         \[\leadsto \color{blue}{-0.25 \cdot \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}} + -1 \cdot \frac{p}{x}} \]

      2. fma-def 44.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{-4 \cdot {p}^{4} + -2 \cdot {p}^{4}}{p \cdot {x}^{3}}, -1 \cdot \frac{p}{x}\right)} \]

      3. distribute-rgt-out 44.8%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{\color{blue}{{p}^{4} \cdot \left(-4 + -2\right)}}{p \cdot {x}^{3}}, -1 \cdot \frac{p}{x}\right) \]

      4. times-frac 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \color{blue}{\frac{{p}^{4}}{p} \cdot \frac{-4 + -2}{{x}^{3}}}, -1 \cdot \frac{p}{x}\right) \]

      5. metadata-eval 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{\color{blue}{-6}}{{x}^{3}}, -1 \cdot \frac{p}{x}\right) \]

      6. associate-*r/ 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{-6}{{x}^{3}}, \color{blue}{\frac{-1 \cdot p}{x}}\right) \]

      7. mul-1-neg 48.1%

         \[\leadsto \mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{-6}{{x}^{3}}, \frac{\color{blue}{-p}}{x}\right) \]

   8. Simplified 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, \frac{{p}^{4}}{p} \cdot \frac{-6}{{x}^{3}}, \frac{-p}{x}\right)} \]

   9. Taylor expanded in p around 0 51.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{p}{x} + 1.5 \cdot \frac{{p}^{3}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. +-commutative 51.2%

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

       2. mul-1-neg 51.2%

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

       3. unsub-neg 51.2%

          \[\leadsto \color{blue}{1.5 \cdot \frac{{p}^{3}}{{x}^{3}} - \frac{p}{x}} \]

       4. *-commutative 51.2%

          \[\leadsto \color{blue}{\frac{{p}^{3}}{{x}^{3}} \cdot 1.5} - \frac{p}{x} \]

       5. cube-div 54.3%

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

   11. Simplified 54.3%

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

   if -0.97999999999999998 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x 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. sqr-neg 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-*l* 100.0%

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

      4. fma-def 100.0%

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

      5. sqr-neg 100.0%

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

      6. fma-def 100.0%

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

      7. associate-*l* 100.0%

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

      8. distribute-lft-in 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. fma-udef 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.98:\\ \;\;\;\;{\left(\frac{p}{x}\right)}^{3} \cdot 1.5 - \frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}\\ \end{array} \]

Alternative 3: 81.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3900:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= x -3900.0)
   (/ (- p) x)
   (sqrt (* 0.5 (+ 1.0 (/ x (+ x (* 2.0 (/ (* p p) x)))))))))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -3900.0) {
		tmp = -p / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * 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 <= (-3900.0d0)) then
        tmp = -p / x
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (x / (x + (2.0d0 * ((p * 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 <= -3900.0) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -3900.0:
		tmp = -p / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))))
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -3900.0)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(x / Float64(x + Float64(2.0 * Float64(Float64(p * p) / x)))))));
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -3900.0)
		tmp = -p / x;
	else
		tmp = sqrt((0.5 * (1.0 + (x / (x + (2.0 * ((p * p) / x)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -3900.0], N[((-p) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(x / N[(x + N[(2.0 * N[(N[(p * p), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3900

   1. Initial program 39.9%

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

      1. sqr-neg 39.9%

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

      2. sqr-neg 39.9%

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

      3. associate-*l* 39.9%

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

      4. fma-def 39.9%

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

      5. sqr-neg 39.9%

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

      6. fma-def 39.9%

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

      7. associate-*l* 39.9%

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

      8. distribute-lft-in 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in x around -inf 45.7%

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

      1. associate-*r/ 45.7%

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

      2. mul-1-neg 45.7%

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

   6. Simplified 45.7%

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

   if -3900 < x

   1. Initial program 92.0%

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

   2. Taylor expanded in p around 0 90.2%

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

      1. unpow2 90.2%

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

   4. Simplified 90.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3900:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{x + 2 \cdot \frac{p \cdot p}{x}}\right)}\\ \end{array} \]

Alternative 4: 67.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2550:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x)
 :precision binary64
 (if (<= x -2550.0) (/ (- p) x) (if (<= x 4.4e-26) (sqrt 0.5) 1.0)))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -2550.0) {
		tmp = -p / x;
	} else if (x <= 4.4e-26) {
		tmp = sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	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 <= (-2550.0d0)) then
        tmp = -p / x
    else if (x <= 4.4d-26) then
        tmp = sqrt(0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -2550.0) {
		tmp = -p / x;
	} else if (x <= 4.4e-26) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -2550.0:
		tmp = -p / x
	elif x <= 4.4e-26:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -2550.0)
		tmp = Float64(Float64(-p) / x);
	elseif (x <= 4.4e-26)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -2550.0)
		tmp = -p / x;
	elseif (x <= 4.4e-26)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -2550.0], N[((-p) / x), $MachinePrecision], If[LessEqual[x, 4.4e-26], N[Sqrt[0.5], $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2550

   1. Initial program 39.9%

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

      1. sqr-neg 39.9%

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

      2. sqr-neg 39.9%

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

      3. associate-*l* 39.9%

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

      4. fma-def 39.9%

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

      5. sqr-neg 39.9%

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

      6. fma-def 39.9%

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

      7. associate-*l* 39.9%

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

      8. distribute-lft-in 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in x around -inf 45.7%

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

      1. associate-*r/ 45.7%

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

      2. mul-1-neg 45.7%

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

   6. Simplified 45.7%

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

   if -2550 < x < 4.4000000000000002e-26

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0 74.9%

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

   if 4.4000000000000002e-26 < 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-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. hypot-def 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. sqrt-prod 100.0%

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

      8. sqrt-prod 57.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. add-log-exp 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 73.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2550:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 55.0% accurate, 35.5× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 (if (<= x -2.85e-136) (/ (- p) x) 1.0))

C

p = abs(p);
double code(double p, double x) {
	double tmp;
	if (x <= -2.85e-136) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	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 <= (-2.85d-136)) then
        tmp = -p / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	double tmp;
	if (x <= -2.85e-136) {
		tmp = -p / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p = abs(p)
def code(p, x):
	tmp = 0
	if x <= -2.85e-136:
		tmp = -p / x
	else:
		tmp = 1.0
	return tmp

Julia

p = abs(p)
function code(p, x)
	tmp = 0.0
	if (x <= -2.85e-136)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p = abs(p)
function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -2.85e-136)
		tmp = -p / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := If[LessEqual[x, -2.85e-136], N[((-p) / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.84999999999999982e-136

   1. Initial program 51.4%

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

      1. sqr-neg 51.4%

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

      2. sqr-neg 51.4%

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

      3. associate-*l* 51.4%

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

      4. fma-def 51.4%

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

      5. sqr-neg 51.4%

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

      6. fma-def 51.4%

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

      7. associate-*l* 51.4%

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

      8. distribute-lft-in 51.4%

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

   3. Simplified 51.4%

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

   4. Taylor expanded in x around -inf 33.0%

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

      1. associate-*r/ 33.0%

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

      2. mul-1-neg 33.0%

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

   6. Simplified 33.0%

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

   if -2.84999999999999982e-136 < 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-log-exp 100.0%

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

      2. +-commutative 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. hypot-def 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. sqrt-prod 100.0%

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

      8. sqrt-prod 52.4%

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

      9. add-sqr-sqrt 100.0%

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

      10. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. add-log-exp 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-lft-in 99.9%

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

      6. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 58.1%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-136}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 35.6% accurate, 215.0× speedup

Math

\[\begin{array}{l} p = |p|\\ \\ 1 \end{array} \]

FPCore

NOTE: p should be positive before calling this function
(FPCore (p x) :precision binary64 1.0)

C

p = abs(p);
double code(double p, double x) {
	return 1.0;
}

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 = 1.0d0
end function

Java

p = Math.abs(p);
public static double code(double p, double x) {
	return 1.0;
}

Python

p = abs(p)
def code(p, x):
	return 1.0

Julia

p = abs(p)
function code(p, x)
	return 1.0
end

MATLAB

p = abs(p)
function tmp = code(p, x)
	tmp = 1.0;
end

Wolfram

NOTE: p should be positive before calling this function
code[p_, x_] := 1.0

Derivation

1. Initial program 78.5%

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

   1. add-log-exp 78.5%

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

   2. +-commutative 78.5%

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

   3. add-sqr-sqrt 78.5%

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

   4. hypot-def 78.5%

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

   5. associate-*r* 78.5%

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

   6. *-commutative 78.5%

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

   7. sqrt-prod 78.5%

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

   8. sqrt-prod 37.4%

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

   9. add-sqr-sqrt 78.5%

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

   10. metadata-eval 78.5%

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

3. Applied egg-rr 78.5%

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

   1. add-log-exp 78.5%

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

   2. *-commutative 78.5%

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

   3. add-log-exp 78.5%

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

   4. *-commutative 78.5%

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

   5. distribute-lft-in 78.5%

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

   6. metadata-eval 78.5%

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

5. Applied egg-rr 78.5%

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

6. Taylor expanded in x around inf 37.5%

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

7. Final simplification 37.5%

   \[\leadsto 1 \]

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 2023293 
(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))))))))