Given's Rotation SVD example

Percentage Accurate: 79.5% → 91.1%

Time: 10.9s

Alternatives: 10

Speedup: 1.0×

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: 79.5% 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: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.8:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left({\left(\frac{p}{x}\right)}^{4}, -6, 2 \cdot {\left(\frac{p}{x}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \log \left(0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.8)
   (sqrt (* 0.5 (fma (pow (/ p x) 4.0) -6.0 (* 2.0 (pow (/ p x) 2.0)))))
   (exp (* 0.5 (log (+ 0.5 (/ 0.5 (/ (hypot x (* p 2.0)) x))))))))

C

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.8) {
		tmp = sqrt((0.5 * fma(pow((p / x), 4.0), -6.0, (2.0 * pow((p / x), 2.0)))));
	} else {
		tmp = exp((0.5 * log((0.5 + (0.5 / (hypot(x, (p * 2.0)) / x))))));
	}
	return tmp;
}

Julia

function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.8)
		tmp = sqrt(Float64(0.5 * fma((Float64(p / x) ^ 4.0), -6.0, Float64(2.0 * (Float64(p / x) ^ 2.0)))));
	else
		tmp = exp(Float64(0.5 * log(Float64(0.5 + Float64(0.5 / Float64(hypot(x, Float64(p * 2.0)) / x))))));
	end
	return tmp
end

Wolfram

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], -0.8], N[Sqrt[N[(0.5 * N[(N[Power[N[(p / x), $MachinePrecision], 4.0], $MachinePrecision] * -6.0 + N[(2.0 * N[Power[N[(p / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(0.5 * N[Log[N[(0.5 + N[(0.5 / N[(N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $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)))) < -0.80000000000000004

   1. Initial program 15.8%

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

      1. add-log-exp 15.9%

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

      2. +-commutative 15.9%

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

      3. add-sqr-sqrt 15.9%

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

      4. hypot-def 15.9%

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

      5. associate-*l* 15.9%

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

      6. sqrt-prod 15.9%

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

      7. metadata-eval 15.9%

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

      8. sqrt-unprod 9.0%

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

      9. add-sqr-sqrt 15.9%

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

   3. Applied egg-rr 15.9%

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

      1. add-log-exp 15.8%

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

      2. clear-num 15.8%

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

   5. Applied egg-rr 15.8%

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

   6. Taylor expanded in x around -inf 44.8%

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

      1. +-commutative 44.8%

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

      2. +-commutative 44.8%

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

      3. distribute-rgt-out 44.8%

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

      4. fma-def 44.8%

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

   8. Simplified 65.9%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\mathsf{fma}\left({\left(\frac{p}{x}\right)}^{4}, -6, 2 \cdot {\left(\frac{p}{x}\right)}^{2}\right)}} \]

   if -0.80000000000000004 < (/.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. pow1/2 100.0%

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

      2. pow-to-exp 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. clear-num 100.0%

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

      6. un-div-inv 100.0%

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

      7. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.8:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left({\left(\frac{p}{x}\right)}^{4}, -6, 2 \cdot {\left(\frac{p}{x}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \log \left(0.5 + \frac{0.5}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}\\ \end{array} \]

Alternative 2: 90.9% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
   (exp (* 0.5 (log (+ 0.5 (/ 0.5 (/ (hypot x (* p 2.0)) x))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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], -1.0], N[Sqrt[N[(0.5 * N[(2.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(0.5 * N[Log[N[(0.5 + N[(0.5 / N[(N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $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)))) < -1

   1. Initial program 14.8%

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

   2. Taylor expanded in x around -inf 48.7%

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

      1. unpow2 48.7%

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

      2. unpow2 48.7%

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

      3. times-frac 65.7%

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

   4. Simplified 65.7%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

      1. pow1/2 99.8%

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

      2. pow-to-exp 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. metadata-eval 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. clear-num 99.8%

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

      6. un-div-inv 99.8%

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

      7. metadata-eval 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 92.5%

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

Alternative 3: 90.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -1.0)
   (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
   (sqrt (* 0.5 (+ 1.0 (/ 1.0 (/ (hypot x (* p 2.0)) x)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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], -1.0], N[Sqrt[N[(0.5 * N[(2.0 * N[(N[(p / x), $MachinePrecision] * N[(p / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[(N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $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)))) < -1

   1. Initial program 14.8%

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

   2. Taylor expanded in x around -inf 48.7%

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

      1. unpow2 48.7%

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

      2. unpow2 48.7%

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

      3. times-frac 65.7%

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

   4. Simplified 65.7%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

   1. Initial program 99.8%

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

      1. add-log-exp 99.8%

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

      2. +-commutative 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. hypot-def 99.8%

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

      5. associate-*l* 99.8%

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

      6. sqrt-prod 99.8%

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

      7. metadata-eval 99.8%

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

      8. sqrt-unprod 45.1%

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

      9. add-sqr-sqrt 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. add-log-exp 99.8%

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

      2. clear-num 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 92.5%

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

Alternative 4: 79.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[p_, x_] := 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. Initial program 81.6%

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

   1. add-sqr-sqrt 81.6%

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

   2. hypot-def 81.6%

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

   3. associate-*l* 81.6%

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

   4. sqrt-prod 81.6%

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

   5. metadata-eval 81.6%

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

   6. sqrt-unprod 37.1%

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

   7. add-sqr-sqrt 81.6%

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

3. Applied egg-rr 81.6%

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

4. Final simplification 81.6%

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

Alternative 5: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-p}{x}\\ \mathbf{if}\;p \leq -2.6 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6.6 \cdot 10^{-232}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x \cdot 0.5}{p}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ (- p) x)))
   (if (<= p -2.6e-62)
     (sqrt 0.5)
     (if (<= p -3.5e-196)
       (/ p x)
       (if (<= p 6.6e-232)
         1.0
         (if (<= p 7e-155)
           t_0
           (if (<= p 3.2e-24)
             1.0
             (if (<= p 2.4e-5)
               t_0
               (sqrt (* 0.5 (+ 1.0 (/ (* x 0.5) p))))))))))))

C

double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= -2.6e-62) {
		tmp = sqrt(0.5);
	} else if (p <= -3.5e-196) {
		tmp = p / x;
	} else if (p <= 6.6e-232) {
		tmp = 1.0;
	} else if (p <= 7e-155) {
		tmp = t_0;
	} else if (p <= 3.2e-24) {
		tmp = 1.0;
	} else if (p <= 2.4e-5) {
		tmp = t_0;
	} else {
		tmp = sqrt((0.5 * (1.0 + ((x * 0.5) / p))));
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -p / x
    if (p <= (-2.6d-62)) then
        tmp = sqrt(0.5d0)
    else if (p <= (-3.5d-196)) then
        tmp = p / x
    else if (p <= 6.6d-232) then
        tmp = 1.0d0
    else if (p <= 7d-155) then
        tmp = t_0
    else if (p <= 3.2d-24) then
        tmp = 1.0d0
    else if (p <= 2.4d-5) then
        tmp = t_0
    else
        tmp = sqrt((0.5d0 * (1.0d0 + ((x * 0.5d0) / p))))
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double t_0 = -p / x;
	double tmp;
	if (p <= -2.6e-62) {
		tmp = Math.sqrt(0.5);
	} else if (p <= -3.5e-196) {
		tmp = p / x;
	} else if (p <= 6.6e-232) {
		tmp = 1.0;
	} else if (p <= 7e-155) {
		tmp = t_0;
	} else if (p <= 3.2e-24) {
		tmp = 1.0;
	} else if (p <= 2.4e-5) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + ((x * 0.5) / p))));
	}
	return tmp;
}

Python

def code(p, x):
	t_0 = -p / x
	tmp = 0
	if p <= -2.6e-62:
		tmp = math.sqrt(0.5)
	elif p <= -3.5e-196:
		tmp = p / x
	elif p <= 6.6e-232:
		tmp = 1.0
	elif p <= 7e-155:
		tmp = t_0
	elif p <= 3.2e-24:
		tmp = 1.0
	elif p <= 2.4e-5:
		tmp = t_0
	else:
		tmp = math.sqrt((0.5 * (1.0 + ((x * 0.5) / p))))
	return tmp

Julia

function code(p, x)
	t_0 = Float64(Float64(-p) / x)
	tmp = 0.0
	if (p <= -2.6e-62)
		tmp = sqrt(0.5);
	elseif (p <= -3.5e-196)
		tmp = Float64(p / x);
	elseif (p <= 6.6e-232)
		tmp = 1.0;
	elseif (p <= 7e-155)
		tmp = t_0;
	elseif (p <= 3.2e-24)
		tmp = 1.0;
	elseif (p <= 2.4e-5)
		tmp = t_0;
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(Float64(x * 0.5) / p))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	t_0 = -p / x;
	tmp = 0.0;
	if (p <= -2.6e-62)
		tmp = sqrt(0.5);
	elseif (p <= -3.5e-196)
		tmp = p / x;
	elseif (p <= 6.6e-232)
		tmp = 1.0;
	elseif (p <= 7e-155)
		tmp = t_0;
	elseif (p <= 3.2e-24)
		tmp = 1.0;
	elseif (p <= 2.4e-5)
		tmp = t_0;
	else
		tmp = sqrt((0.5 * (1.0 + ((x * 0.5) / p))));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := Block[{t$95$0 = N[((-p) / x), $MachinePrecision]}, If[LessEqual[p, -2.6e-62], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, -3.5e-196], N[(p / x), $MachinePrecision], If[LessEqual[p, 6.6e-232], 1.0, If[LessEqual[p, 7e-155], t$95$0, If[LessEqual[p, 3.2e-24], 1.0, If[LessEqual[p, 2.4e-5], t$95$0, N[Sqrt[N[(0.5 * N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] / p), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if p < -2.5999999999999999e-62

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 83.4%

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

   if -2.5999999999999999e-62 < p < -3.50000000000000004e-196

   1. Initial program 41.3%

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

   2. Taylor expanded in x around -inf 29.6%

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

      1. unpow2 29.6%

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

      2. unpow2 29.6%

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

      3. times-frac 41.4%

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

   4. Simplified 41.4%

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

   5. Taylor expanded in p around 0 64.4%

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

   if -3.50000000000000004e-196 < p < 6.5999999999999997e-232 or 7.00000000000000031e-155 < p < 3.20000000000000012e-24

   1. Initial program 78.6%

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

   2. Taylor expanded in x around inf 64.2%

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

   if 6.5999999999999997e-232 < p < 7.00000000000000031e-155 or 3.20000000000000012e-24 < p < 2.4000000000000001e-5

   1. Initial program 24.1%

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

      1. pow1/2 24.1%

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

      2. pow-to-exp 24.1%

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

   3. Applied egg-rr 24.1%

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

   4. Taylor expanded in x around -inf 81.8%

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

      1. associate-*r/ 81.8%

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

      2. neg-mul-1 81.8%

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

   6. Simplified 81.8%

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

   if 2.4000000000000001e-5 < p

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-*l/ 96.1%

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

   4. Simplified 96.1%

      \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\frac{x \cdot 0.5}{p}}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -2.6 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -3.5 \cdot 10^{-196}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 6.6 \cdot 10^{-232}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 7 \cdot 10^{-155}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x \cdot 0.5}{p}\right)}\\ \end{array} \]

Alternative 6: 76.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+58}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{1 + 2 \cdot \left(p \cdot \frac{p}{x \cdot x}\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= x -2.15e+58)
   (/ p x)
   (sqrt (* 0.5 (+ 1.0 (/ 1.0 (+ 1.0 (* 2.0 (* p (/ p (* x x)))))))))))

C

double code(double p, double x) {
	double tmp;
	if (x <= -2.15e+58) {
		tmp = p / x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (1.0 / (1.0 + (2.0 * (p * (p / (x * x)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.15d+58)) then
        tmp = p / x
    else
        tmp = sqrt((0.5d0 * (1.0d0 + (1.0d0 / (1.0d0 + (2.0d0 * (p * (p / (x * x)))))))))
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (x <= -2.15e+58) {
		tmp = p / x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (1.0 / (1.0 + (2.0 * (p * (p / (x * x)))))))));
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if x <= -2.15e+58:
		tmp = p / x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (1.0 / (1.0 + (2.0 * (p * (p / (x * x)))))))))
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (x <= -2.15e+58)
		tmp = Float64(p / x);
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(p * Float64(p / Float64(x * x)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (x <= -2.15e+58)
		tmp = p / x;
	else
		tmp = sqrt((0.5 * (1.0 + (1.0 / (1.0 + (2.0 * (p * (p / (x * x)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[x, -2.15e+58], N[(p / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[(1.0 + N[(2.0 * N[(p * N[(p / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.14999999999999996e58

   1. Initial program 46.7%

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

   2. Taylor expanded in x around -inf 40.3%

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

      1. unpow2 40.3%

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

      2. unpow2 40.3%

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

      3. times-frac 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in p around 0 52.9%

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

   if -2.14999999999999996e58 < x

   1. Initial program 88.0%

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

      1. add-log-exp 88.0%

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

      2. +-commutative 88.0%

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

      3. add-sqr-sqrt 88.0%

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

      4. hypot-def 88.0%

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

      5. associate-*l* 88.0%

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

      6. sqrt-prod 88.0%

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

      7. metadata-eval 88.0%

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

      8. sqrt-unprod 41.0%

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

      9. add-sqr-sqrt 88.0%

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

   3. Applied egg-rr 88.0%

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

      1. add-log-exp 88.0%

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

      2. clear-num 88.0%

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

   5. Applied egg-rr 88.0%

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

   6. Taylor expanded in x around inf 85.7%

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

      1. unpow2 85.7%

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

      2. associate-*r/ 85.7%

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

      3. unpow2 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+58}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{1 + 2 \cdot \left(p \cdot \frac{p}{x \cdot x}\right)}\right)}\\ \end{array} \]

Alternative 7: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -2.8 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.25 \cdot 10^{-195}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 4.95 \cdot 10^{-231}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.06 \cdot 10^{-151}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4.2 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p -2.8e-63)
   (sqrt 0.5)
   (if (<= p -2.25e-195)
     (/ p x)
     (if (<= p 4.95e-231)
       1.0
       (if (<= p 1.06e-151) (/ (- p) x) (if (<= p 4.2e-62) 1.0 (sqrt 0.5)))))))

C

double code(double p, double x) {
	double tmp;
	if (p <= -2.8e-63) {
		tmp = sqrt(0.5);
	} else if (p <= -2.25e-195) {
		tmp = p / x;
	} else if (p <= 4.95e-231) {
		tmp = 1.0;
	} else if (p <= 1.06e-151) {
		tmp = -p / x;
	} else if (p <= 4.2e-62) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= (-2.8d-63)) then
        tmp = sqrt(0.5d0)
    else if (p <= (-2.25d-195)) then
        tmp = p / x
    else if (p <= 4.95d-231) then
        tmp = 1.0d0
    else if (p <= 1.06d-151) then
        tmp = -p / x
    else if (p <= 4.2d-62) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= -2.8e-63) {
		tmp = Math.sqrt(0.5);
	} else if (p <= -2.25e-195) {
		tmp = p / x;
	} else if (p <= 4.95e-231) {
		tmp = 1.0;
	} else if (p <= 1.06e-151) {
		tmp = -p / x;
	} else if (p <= 4.2e-62) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -2.8e-63:
		tmp = math.sqrt(0.5)
	elif p <= -2.25e-195:
		tmp = p / x
	elif p <= 4.95e-231:
		tmp = 1.0
	elif p <= 1.06e-151:
		tmp = -p / x
	elif p <= 4.2e-62:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -2.8e-63)
		tmp = sqrt(0.5);
	elseif (p <= -2.25e-195)
		tmp = Float64(p / x);
	elseif (p <= 4.95e-231)
		tmp = 1.0;
	elseif (p <= 1.06e-151)
		tmp = Float64(Float64(-p) / x);
	elseif (p <= 4.2e-62)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -2.8e-63)
		tmp = sqrt(0.5);
	elseif (p <= -2.25e-195)
		tmp = p / x;
	elseif (p <= 4.95e-231)
		tmp = 1.0;
	elseif (p <= 1.06e-151)
		tmp = -p / x;
	elseif (p <= 4.2e-62)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -2.8e-63], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, -2.25e-195], N[(p / x), $MachinePrecision], If[LessEqual[p, 4.95e-231], 1.0, If[LessEqual[p, 1.06e-151], N[((-p) / x), $MachinePrecision], If[LessEqual[p, 4.2e-62], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < -2.8000000000000002e-63 or 4.1999999999999998e-62 < p

   1. Initial program 93.8%

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

   2. Taylor expanded in x around 0 83.8%

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

   if -2.8000000000000002e-63 < p < -2.25e-195

   1. Initial program 41.3%

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

   2. Taylor expanded in x around -inf 29.6%

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

      1. unpow2 29.6%

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

      2. unpow2 29.6%

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

      3. times-frac 41.4%

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

   4. Simplified 41.4%

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

   5. Taylor expanded in p around 0 64.4%

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

   if -2.25e-195 < p < 4.9500000000000001e-231 or 1.06e-151 < p < 4.1999999999999998e-62

   1. Initial program 78.6%

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

   2. Taylor expanded in x around inf 67.7%

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

   if 4.9500000000000001e-231 < p < 1.06e-151

   1. Initial program 30.9%

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

      1. pow1/2 30.9%

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

      2. pow-to-exp 30.9%

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

   3. Applied egg-rr 30.9%

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

   4. Taylor expanded in x around -inf 75.7%

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

      1. associate-*r/ 75.7%

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

      2. neg-mul-1 75.7%

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

   6. Simplified 75.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -2.8 \cdot 10^{-63}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -2.25 \cdot 10^{-195}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 4.95 \cdot 10^{-231}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 1.06 \cdot 10^{-151}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 4.2 \cdot 10^{-62}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 8: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -6.2 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.85 \cdot 10^{-278}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p -6.2e-65)
   (sqrt 0.5)
   (if (<= p -1.85e-278) (/ p x) (if (<= p 5e-10) (/ (- p) x) (sqrt 0.5)))))

C

double code(double p, double x) {
	double tmp;
	if (p <= -6.2e-65) {
		tmp = sqrt(0.5);
	} else if (p <= -1.85e-278) {
		tmp = p / x;
	} else if (p <= 5e-10) {
		tmp = -p / x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= (-6.2d-65)) then
        tmp = sqrt(0.5d0)
    else if (p <= (-1.85d-278)) then
        tmp = p / x
    else if (p <= 5d-10) then
        tmp = -p / x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= -6.2e-65) {
		tmp = Math.sqrt(0.5);
	} else if (p <= -1.85e-278) {
		tmp = p / x;
	} else if (p <= 5e-10) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -6.2e-65:
		tmp = math.sqrt(0.5)
	elif p <= -1.85e-278:
		tmp = p / x
	elif p <= 5e-10:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -6.2e-65)
		tmp = sqrt(0.5);
	elseif (p <= -1.85e-278)
		tmp = Float64(p / x);
	elseif (p <= 5e-10)
		tmp = Float64(Float64(-p) / x);
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -6.2e-65)
		tmp = sqrt(0.5);
	elseif (p <= -1.85e-278)
		tmp = p / x;
	elseif (p <= 5e-10)
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -6.2e-65], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, -1.85e-278], N[(p / x), $MachinePrecision], If[LessEqual[p, 5e-10], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -6.20000000000000032e-65 or 5.00000000000000031e-10 < p

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 87.8%

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

   if -6.20000000000000032e-65 < p < -1.85000000000000011e-278

   1. Initial program 54.4%

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

   2. Taylor expanded in x around -inf 19.6%

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

      1. unpow2 19.6%

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

      2. unpow2 19.6%

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

      3. times-frac 32.8%

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

   4. Simplified 32.8%

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

   5. Taylor expanded in p around 0 50.4%

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

   if -1.85000000000000011e-278 < p < 5.00000000000000031e-10

   1. Initial program 64.9%

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

      1. pow1/2 64.9%

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

      2. pow-to-exp 64.9%

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

   3. Applied egg-rr 64.9%

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

   4. Taylor expanded in x around -inf 47.4%

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

      1. associate-*r/ 47.4%

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

      2. neg-mul-1 47.4%

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

   6. Simplified 47.4%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -6.2 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -1.85 \cdot 10^{-278}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 9: 26.1% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -1.85 \cdot 10^{-278}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= p -1.85e-278) (/ p x) (/ (- p) x)))

C

double code(double p, double x) {
	double tmp;
	if (p <= -1.85e-278) {
		tmp = p / x;
	} else {
		tmp = -p / x;
	}
	return tmp;
}

Fortran

real(8) function code(p, x)
    real(8), intent (in) :: p
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p <= (-1.85d-278)) then
        tmp = p / x
    else
        tmp = -p / x
    end if
    code = tmp
end function

Java

public static double code(double p, double x) {
	double tmp;
	if (p <= -1.85e-278) {
		tmp = p / x;
	} else {
		tmp = -p / x;
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -1.85e-278:
		tmp = p / x
	else:
		tmp = -p / x
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -1.85e-278)
		tmp = Float64(p / x);
	else
		tmp = Float64(Float64(-p) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -1.85e-278)
		tmp = p / x;
	else
		tmp = -p / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -1.85e-278], N[(p / x), $MachinePrecision], N[((-p) / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -1.85000000000000011e-278

   1. Initial program 80.3%

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

   2. Taylor expanded in x around -inf 13.5%

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

      1. unpow2 13.5%

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

      2. unpow2 13.5%

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

      3. times-frac 18.6%

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

   4. Simplified 18.6%

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

   5. Taylor expanded in p around 0 23.5%

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

   if -1.85000000000000011e-278 < p

   1. Initial program 82.9%

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

      1. pow1/2 82.9%

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

      2. pow-to-exp 82.9%

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

   3. Applied egg-rr 82.9%

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

   4. Taylor expanded in x around -inf 24.5%

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

      1. associate-*r/ 24.5%

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

      2. neg-mul-1 24.5%

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

   6. Simplified 24.5%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -1.85 \cdot 10^{-278}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 10: 17.1% accurate, 71.7× speedup

Math

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

FPCore

(FPCore (p x) :precision binary64 (/ p x))

C

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

Fortran

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

Java

public static double code(double p, double x) {
	return p / x;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[p_, x_] := N[(p / x), $MachinePrecision]

Derivation

1. Initial program 81.6%

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

2. Taylor expanded in x around -inf 14.5%

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

   1. unpow2 14.5%

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

   2. unpow2 14.5%

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

   3. times-frac 18.4%

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

4. Simplified 18.4%

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

5. Taylor expanded in p around 0 15.9%

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

6. Final simplification 15.9%

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

Developer target: 79.5% 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 2023194 
(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))))))))