Given's Rotation SVD example

Percentage Accurate: 79.3% → 91.5%

Time: 9.1s

Alternatives: 7

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\sqrt{{\left(\frac{x}{p}\right)}^{-2} + 0.5 \cdot \left(-6 \cdot {\left(\frac{p}{x}\right)}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(1 + \left(\frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)} + -1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.5)
   (sqrt (+ (pow (/ x p) -2.0) (* 0.5 (* -6.0 (pow (/ p x) 4.0)))))
   (sqrt (* 0.5 (+ 1.0 (+ 1.0 (+ (/ x (hypot (* p 2.0) x)) -1.0)))))))

C

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5) {
		tmp = sqrt((pow((x / p), -2.0) + (0.5 * (-6.0 * pow((p / x), 4.0)))));
	} else {
		tmp = sqrt((0.5 * (1.0 + (1.0 + ((x / hypot((p * 2.0), x)) + -1.0)))));
	}
	return tmp;
}

Java

public static double code(double p, double x) {
	double tmp;
	if ((x / Math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5) {
		tmp = Math.sqrt((Math.pow((x / p), -2.0) + (0.5 * (-6.0 * Math.pow((p / x), 4.0)))));
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (1.0 + ((x / Math.hypot((p * 2.0), x)) + -1.0)))));
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5:
		tmp = math.sqrt((math.pow((x / p), -2.0) + (0.5 * (-6.0 * math.pow((p / x), 4.0)))))
	else:
		tmp = math.sqrt((0.5 * (1.0 + (1.0 + ((x / math.hypot((p * 2.0), x)) + -1.0)))))
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.5)
		tmp = sqrt(Float64((Float64(x / p) ^ -2.0) + Float64(0.5 * Float64(-6.0 * (Float64(p / x) ^ 4.0)))));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 + Float64(Float64(x / hypot(Float64(p * 2.0), x)) + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.5)
		tmp = sqrt((((x / p) ^ -2.0) + (0.5 * (-6.0 * ((p / x) ^ 4.0)))));
	else
		tmp = sqrt((0.5 * (1.0 + (1.0 + ((x / hypot((p * 2.0), x)) + -1.0)))));
	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], -0.5], N[Sqrt[N[(N[Power[N[(x / p), $MachinePrecision], -2.0], $MachinePrecision] + N[(0.5 * N[(-6.0 * N[Power[N[(p / x), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 + N[(N[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] + -1.0), $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)))) < -0.5

   1. Initial program 13.1%

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

      1. add-sqr-sqrt 13.1%

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

      2. hypot-def 13.2%

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

      3. associate-*l* 13.2%

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

      4. sqrt-prod 13.2%

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

      5. metadata-eval 13.2%

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

      6. sqrt-unprod 8.5%

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

      7. add-sqr-sqrt 13.2%

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

   3. Applied egg-rr 13.2%

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

   4. Taylor expanded in x around -inf 39.7%

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

      1. fma-def 39.7%

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

      2. unpow2 39.7%

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

      3. unpow2 39.7%

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

      4. distribute-rgt-out 39.7%

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

      5. metadata-eval 39.7%

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

   6. Simplified 39.7%

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

      1. expm1-log1p-u 39.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \mathsf{fma}\left(2, \frac{p \cdot p}{x \cdot x}, \frac{{p}^{4}}{{x}^{4}} \cdot -6\right)}\right)\right)} \]

      2. expm1-udef 12.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \mathsf{fma}\left(2, \frac{p \cdot p}{x \cdot x}, \frac{{p}^{4}}{{x}^{4}} \cdot -6\right)}\right)} - 1} \]

   8. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{0.5 \cdot \mathsf{fma}\left(2, {\left(\frac{p}{x}\right)}^{2}, {\left(\frac{p}{x}\right)}^{4} \cdot -6\right)}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 66.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5 \cdot \mathsf{fma}\left(2, {\left(\frac{p}{x}\right)}^{2}, {\left(\frac{p}{x}\right)}^{4} \cdot -6\right)}\right)\right)} \]

      2. expm1-log1p 66.4%

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

      3. fma-udef 66.4%

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

      4. unpow2 66.4%

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

      5. times-frac 53.3%

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

      6. unpow2 53.3%

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

      7. unpow2 53.3%

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

      8. distribute-lft-in 53.3%

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

   10. Simplified 66.1%

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

   if -0.5 < (/.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-sqr-sqrt 100.0%

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

      2. hypot-def 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqrt-prod 100.0%

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

      5. metadata-eval 100.0%

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

      6. sqrt-unprod 46.9%

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

      7. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)\right)}\right)} \]

      2. expm1-udef 100.0%

         \[\leadsto \sqrt{0.5 \cdot \left(1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(2 \cdot p, x\right)}\right)} - 1\right)}\right)} \]

      3. log1p-udef 100.0%

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

      4. add-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate--l+ 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\sqrt{{\left(\frac{x}{p}\right)}^{-2} + 0.5 \cdot \left(-6 \cdot {\left(\frac{p}{x}\right)}^{4}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \left(1 + \left(\frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)} + -1\right)\right)\right)}\\ \end{array} \]

Alternative 2: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999996:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9999996)
   (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
   (sqrt (* 0.5 (fma (/ 1.0 (hypot x (* p 2.0))) x 1.0)))))

C

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999996) {
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * fma((1.0 / hypot(x, (p * 2.0))), x, 1.0)));
	}
	return tmp;
}

Julia

function code(p, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p * Float64(4.0 * p)) + Float64(x * x)))) <= -0.9999996)
		tmp = sqrt(Float64(0.5 * Float64(2.0 * Float64(Float64(p / x) * Float64(p / x)))));
	else
		tmp = sqrt(Float64(0.5 * fma(Float64(1.0 / hypot(x, Float64(p * 2.0))), x, 1.0)));
	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], -0.9999996], 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[(N[(1.0 / N[Sqrt[x ^ 2 + N[(p * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * x + 1.0), $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)))) < -0.99999959999999999

   1. Initial program 11.8%

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

   2. Taylor expanded in x around -inf 52.7%

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

      1. unpow2 52.7%

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

      2. unpow2 52.7%

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

      3. times-frac 66.0%

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

   4. Simplified 66.0%

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

   if -0.99999959999999999 < (/.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. +-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. associate-/r/ 99.8%

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

      4. fma-def 99.8%

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

      5. +-commutative 99.8%

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

      6. add-sqr-sqrt 99.8%

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

      7. hypot-def 99.9%

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

      8. associate-*l* 99.9%

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

      9. sqrt-prod 99.9%

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

      10. metadata-eval 99.9%

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

      11. sqrt-unprod 47.0%

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

      12. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999996:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(x, p \cdot 2\right)}, x, 1\right)}\\ \end{array} \]

Alternative 3: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999996:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9999996)
   (sqrt (* 0.5 (* 2.0 (* (/ p x) (/ p x)))))
   (sqrt (* 0.5 (+ 1.0 (/ x (hypot (* p 2.0) x)))))))

C

double code(double p, double x) {
	double tmp;
	if ((x / sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999996) {
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = sqrt((0.5 * (1.0 + (x / hypot((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)))) <= -0.9999996) {
		tmp = Math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (x / Math.hypot((p * 2.0), x)))));
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if (x / math.sqrt(((p * (4.0 * p)) + (x * x)))) <= -0.9999996:
		tmp = math.sqrt((0.5 * (2.0 * ((p / x) * (p / x)))))
	else:
		tmp = math.sqrt((0.5 * (1.0 + (x / math.hypot((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)))) <= -0.9999996)
		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(x / hypot(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)))) <= -0.9999996)
		tmp = sqrt((0.5 * (2.0 * ((p / x) * (p / x)))));
	else
		tmp = sqrt((0.5 * (1.0 + (x / hypot((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], -0.9999996], 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[(x / N[Sqrt[N[(p * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99999959999999999

   1. Initial program 11.8%

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

   2. Taylor expanded in x around -inf 52.7%

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

      1. unpow2 52.7%

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

      2. unpow2 52.7%

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

      3. times-frac 66.0%

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

   4. Simplified 66.0%

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

   if -0.99999959999999999 < (/.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-sqr-sqrt 99.8%

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

      2. hypot-def 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqrt-prod 99.9%

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

      5. metadata-eval 99.9%

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

      6. sqrt-unprod 47.0%

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

      7. add-sqr-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999996:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot \left(\frac{p}{x} \cdot \frac{p}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{x}{\mathsf{hypot}\left(p \cdot 2, x\right)}\right)}\\ \end{array} \]

Alternative 4: 66.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -5.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8 \cdot 10^{-221}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p -5.8e-65)
   (sqrt 0.5)
   (if (<= p -8e-221) (/ p x) (if (<= p 1.5e-74) (/ (- p) x) (sqrt 0.5)))))

C

double code(double p, double x) {
	double tmp;
	if (p <= -5.8e-65) {
		tmp = sqrt(0.5);
	} else if (p <= -8e-221) {
		tmp = p / x;
	} else if (p <= 1.5e-74) {
		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 <= (-5.8d-65)) then
        tmp = sqrt(0.5d0)
    else if (p <= (-8d-221)) then
        tmp = p / x
    else if (p <= 1.5d-74) 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 <= -5.8e-65) {
		tmp = Math.sqrt(0.5);
	} else if (p <= -8e-221) {
		tmp = p / x;
	} else if (p <= 1.5e-74) {
		tmp = -p / x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -5.8e-65:
		tmp = math.sqrt(0.5)
	elif p <= -8e-221:
		tmp = p / x
	elif p <= 1.5e-74:
		tmp = -p / x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -5.8e-65)
		tmp = sqrt(0.5);
	elseif (p <= -8e-221)
		tmp = Float64(p / x);
	elseif (p <= 1.5e-74)
		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 <= -5.8e-65)
		tmp = sqrt(0.5);
	elseif (p <= -8e-221)
		tmp = p / x;
	elseif (p <= 1.5e-74)
		tmp = -p / x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -5.8e-65], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, -8e-221], N[(p / x), $MachinePrecision], If[LessEqual[p, 1.5e-74], N[((-p) / x), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if p < -5.7999999999999996e-65 or 1.50000000000000003e-74 < p

   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 x around 0 78.1%

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

   if -5.7999999999999996e-65 < p < -8.00000000000000014e-221

   1. Initial program 59.1%

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

   2. Taylor expanded in x around -inf 9.9%

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

      1. unpow2 9.9%

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

      2. unpow2 9.9%

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

      3. times-frac 18.5%

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

   4. Simplified 18.5%

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

   5. Taylor expanded in p around 0 45.0%

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

   if -8.00000000000000014e-221 < p < 1.50000000000000003e-74

   1. Initial program 70.3%

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

      1. log1p-expm1-u 70.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. log1p-udef 70.3%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      3. +-commutative 70.3%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}}\right)\right) \]

      4. distribute-rgt-in 70.3%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5 + 1 \cdot 0.5}}\right)\right) \]

      5. metadata-eval 70.3%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5 + \color{blue}{0.5}}\right)\right) \]

      6. fma-def 70.3%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}}\right)\right) \]

   3. Applied egg-rr 70.3%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)}\right)\right)} \]

   4. Taylor expanded in x around -inf 39.2%

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

      1. associate-*r/ 39.2%

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

      2. mul-1-neg 39.2%

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

   6. Simplified 39.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -5.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq -8 \cdot 10^{-221}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;p \leq 1.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 5: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -1.25 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4300000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x)
 :precision binary64
 (if (<= p -1.25e-42) (sqrt 0.5) (if (<= p 4300000000000.0) 1.0 (sqrt 0.5))))

C

double code(double p, double x) {
	double tmp;
	if (p <= -1.25e-42) {
		tmp = sqrt(0.5);
	} else if (p <= 4300000000000.0) {
		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 <= (-1.25d-42)) then
        tmp = sqrt(0.5d0)
    else if (p <= 4300000000000.0d0) 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 <= -1.25e-42) {
		tmp = Math.sqrt(0.5);
	} else if (p <= 4300000000000.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -1.25e-42:
		tmp = math.sqrt(0.5)
	elif p <= 4300000000000.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -1.25e-42)
		tmp = sqrt(0.5);
	elseif (p <= 4300000000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(p, x)
	tmp = 0.0;
	if (p <= -1.25e-42)
		tmp = sqrt(0.5);
	elseif (p <= 4300000000000.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -1.25e-42], N[Sqrt[0.5], $MachinePrecision], If[LessEqual[p, 4300000000000.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if p < -1.25000000000000001e-42 or 4.3e12 < p

   1. Initial program 92.6%

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

   2. Taylor expanded in x around 0 84.8%

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

   if -1.25000000000000001e-42 < p < 4.3e12

   1. Initial program 72.0%

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

   2. Taylor expanded in x around inf 62.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -1.25 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{elif}\;p \leq 4300000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]

Alternative 6: 24.6% accurate, 35.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;p \leq -8 \cdot 10^{-221}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (p x) :precision binary64 (if (<= p -8e-221) (/ p x) (/ (- p) x)))

C

double code(double p, double x) {
	double tmp;
	if (p <= -8e-221) {
		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 <= (-8d-221)) 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 <= -8e-221) {
		tmp = p / x;
	} else {
		tmp = -p / x;
	}
	return tmp;
}

Python

def code(p, x):
	tmp = 0
	if p <= -8e-221:
		tmp = p / x
	else:
		tmp = -p / x
	return tmp

Julia

function code(p, x)
	tmp = 0.0
	if (p <= -8e-221)
		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 <= -8e-221)
		tmp = p / x;
	else
		tmp = -p / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[p_, x_] := If[LessEqual[p, -8e-221], N[(p / x), $MachinePrecision], N[((-p) / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < -8.00000000000000014e-221

   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 -inf 12.8%

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

      1. unpow2 12.8%

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

      2. unpow2 12.8%

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

      3. times-frac 14.6%

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

   4. Simplified 14.6%

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

   5. Taylor expanded in p around 0 18.1%

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

   if -8.00000000000000014e-221 < p

   1. Initial program 84.3%

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

      1. log1p-expm1-u 84.3%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      2. log1p-udef 84.2%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)\right)} \]

      3. +-commutative 84.2%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{0.5 \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}}\right)\right) \]

      4. distribute-rgt-in 84.2%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5 + 1 \cdot 0.5}}\right)\right) \]

      5. metadata-eval 84.2%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot 0.5 + \color{blue}{0.5}}\right)\right) \]

      6. fma-def 84.2%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, 0.5, 0.5\right)}}\right)\right) \]

   3. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt{\mathsf{fma}\left(\frac{x}{\mathsf{hypot}\left(x, 2 \cdot p\right)}, 0.5, 0.5\right)}\right)\right)} \]

   4. Taylor expanded in x around -inf 21.3%

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

      1. associate-*r/ 21.3%

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

      2. mul-1-neg 21.3%

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

   6. Simplified 21.3%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq -8 \cdot 10^{-221}:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-p}{x}\\ \end{array} \]

Alternative 7: 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 84.7%

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

2. Taylor expanded in x around -inf 13.3%

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

   1. unpow2 13.3%

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

   2. unpow2 13.3%

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

   3. times-frac 15.8%

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

4. Simplified 15.8%

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

5. Taylor expanded in p around 0 11.1%

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

6. Final simplification 11.1%

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

Developer target: 79.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023193 
(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))))))))