Given's Rotation SVD example

Percentage Accurate: 79.3% → 99.7%

Time: 8.4s

Alternatives: 7

Speedup: 0.7×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))) -1.0)
   (* -1.0 (/ p_m x))
   (sqrt (* 0.5 (log (exp (+ 1.0 (/ x (hypot x (* 2.0 p_m))))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(-1.0 * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(1.0 + N[(x / N[Sqrt[x ^ 2 + N[(2.0 * p$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.3%

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

      1. +-commutative 18.3%

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

      2. distribute-lft-in 18.3%

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

      3. associate-*r/ 18.3%

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

      4. +-commutative 18.3%

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

      5. add-sqr-sqrt 18.3%

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

      6. hypot-define 18.3%

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

      7. associate-*l* 18.3%

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

      8. sqrt-prod 18.3%

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

      9. metadata-eval 18.3%

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

      10. sqrt-unprod 7.1%

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

      11. add-sqr-sqrt 18.3%

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

      12. metadata-eval 18.3%

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

   4. Applied egg-rr 18.3%

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

   5. Taylor expanded in x around -inf 61.4%

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

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

   1. Initial program 99.7%

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

      1. add-log-exp 99.7%

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

      2. +-commutative 99.7%

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

      3. add-sqr-sqrt 99.7%

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

      4. hypot-define 99.7%

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

      5. associate-*l* 99.7%

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

      6. sqrt-prod 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-unprod 45.0%

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

      9. add-sqr-sqrt 99.7%

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

   4. Applied egg-rr 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))) -1.0)
   (* -1.0 (/ p_m x))
   (exp (* (log (fma (/ x (hypot x (* 2.0 p_m))) 0.5 0.5)) 0.5))))

C

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

Julia

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(-1.0 * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[N[(N[(x / N[Sqrt[x ^ 2 + N[(2.0 * p$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] * 0.5), $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 18.3%

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

      1. +-commutative 18.3%

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

      2. distribute-lft-in 18.3%

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

      3. associate-*r/ 18.3%

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

      4. +-commutative 18.3%

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

      5. add-sqr-sqrt 18.3%

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

      6. hypot-define 18.3%

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

      7. associate-*l* 18.3%

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

      8. sqrt-prod 18.3%

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

      9. metadata-eval 18.3%

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

      10. sqrt-unprod 7.1%

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

      11. add-sqr-sqrt 18.3%

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

      12. metadata-eval 18.3%

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

   4. Applied egg-rr 18.3%

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

   5. Taylor expanded in x around -inf 61.4%

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

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

   1. Initial program 99.7%

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

      1. pow1/2 99.7%

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

      2. pow-to-exp 99.7%

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

   4. Applied egg-rr 99.7%

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

Alternative 3: 99.7% accurate, 0.7× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* (* 4.0 p_m) p_m) (* x x)))) -1.0)
   (* -1.0 (/ p_m x))
   (sqrt (+ (/ (* 0.5 x) (hypot x (* 2.0 p_m))) 0.5))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(N[(4.0 * p$95$m), $MachinePrecision] * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[(-1.0 * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(0.5 * x), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(2.0 * p$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $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 18.3%

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

      1. +-commutative 18.3%

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

      2. distribute-lft-in 18.3%

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

      3. associate-*r/ 18.3%

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

      4. +-commutative 18.3%

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

      5. add-sqr-sqrt 18.3%

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

      6. hypot-define 18.3%

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

      7. associate-*l* 18.3%

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

      8. sqrt-prod 18.3%

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

      9. metadata-eval 18.3%

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

      10. sqrt-unprod 7.1%

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

      11. add-sqr-sqrt 18.3%

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

      12. metadata-eval 18.3%

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

   4. Applied egg-rr 18.3%

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

   5. Taylor expanded in x around -inf 61.4%

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

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

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. associate-*r/ 99.7%

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

      4. +-commutative 99.7%

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

      5. add-sqr-sqrt 99.7%

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

      6. hypot-define 99.7%

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

      7. associate-*l* 99.7%

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

      8. sqrt-prod 99.7%

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

      9. metadata-eval 99.7%

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

      10. sqrt-unprod 45.0%

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

      11. add-sqr-sqrt 99.7%

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

      12. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

Alternative 4: 67.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.15 \cdot 10^{-138}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 4.9 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{p\_m}{x}\\ \mathbf{elif}\;p\_m \leq 5.4 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 3.3 \cdot 10^{+37}:\\ \;\;\;\;\sqrt{0.5 + 0.25 \cdot \frac{x}{p\_m}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.15e-138)
   1.0
   (if (<= p_m 4.9e-82)
     (* -1.0 (/ p_m x))
     (if (<= p_m 5.4e-6)
       1.0
       (if (<= p_m 3.3e+37) (sqrt (+ 0.5 (* 0.25 (/ x p_m)))) (sqrt 0.5))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.15e-138) {
		tmp = 1.0;
	} else if (p_m <= 4.9e-82) {
		tmp = -1.0 * (p_m / x);
	} else if (p_m <= 5.4e-6) {
		tmp = 1.0;
	} else if (p_m <= 3.3e+37) {
		tmp = sqrt((0.5 + (0.25 * (x / p_m))));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.15d-138) then
        tmp = 1.0d0
    else if (p_m <= 4.9d-82) then
        tmp = (-1.0d0) * (p_m / x)
    else if (p_m <= 5.4d-6) then
        tmp = 1.0d0
    else if (p_m <= 3.3d+37) then
        tmp = sqrt((0.5d0 + (0.25d0 * (x / p_m))))
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.15e-138) {
		tmp = 1.0;
	} else if (p_m <= 4.9e-82) {
		tmp = -1.0 * (p_m / x);
	} else if (p_m <= 5.4e-6) {
		tmp = 1.0;
	} else if (p_m <= 3.3e+37) {
		tmp = Math.sqrt((0.5 + (0.25 * (x / p_m))));
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.15e-138:
		tmp = 1.0
	elif p_m <= 4.9e-82:
		tmp = -1.0 * (p_m / x)
	elif p_m <= 5.4e-6:
		tmp = 1.0
	elif p_m <= 3.3e+37:
		tmp = math.sqrt((0.5 + (0.25 * (x / p_m))))
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.15e-138)
		tmp = 1.0;
	elseif (p_m <= 4.9e-82)
		tmp = Float64(-1.0 * Float64(p_m / x));
	elseif (p_m <= 5.4e-6)
		tmp = 1.0;
	elseif (p_m <= 3.3e+37)
		tmp = sqrt(Float64(0.5 + Float64(0.25 * Float64(x / p_m))));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.15e-138)
		tmp = 1.0;
	elseif (p_m <= 4.9e-82)
		tmp = -1.0 * (p_m / x);
	elseif (p_m <= 5.4e-6)
		tmp = 1.0;
	elseif (p_m <= 3.3e+37)
		tmp = sqrt((0.5 + (0.25 * (x / p_m))));
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.15e-138], 1.0, If[LessEqual[p$95$m, 4.9e-82], N[(-1.0 * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 5.4e-6], 1.0, If[LessEqual[p$95$m, 3.3e+37], N[Sqrt[N[(0.5 + N[(0.25 * N[(x / p$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 1.14999999999999995e-138 or 4.9000000000000003e-82 < p < 5.39999999999999997e-6

   1. Initial program 80.6%

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

      1. +-commutative 80.6%

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

      2. distribute-lft-in 80.6%

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

      3. associate-*r/ 80.6%

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

      4. +-commutative 80.6%

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

      5. add-sqr-sqrt 80.6%

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

      6. hypot-define 80.6%

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

      7. associate-*l* 80.6%

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

      8. sqrt-prod 80.6%

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

      9. metadata-eval 80.6%

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

      10. sqrt-unprod 17.6%

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

      11. add-sqr-sqrt 80.6%

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

      12. metadata-eval 80.6%

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

   4. Applied egg-rr 80.6%

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

   5. Taylor expanded in x around inf 41.9%

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

   if 1.14999999999999995e-138 < p < 4.9000000000000003e-82

   1. Initial program 47.1%

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

      1. +-commutative 47.1%

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

      2. distribute-lft-in 47.1%

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

      3. associate-*r/ 47.1%

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

      4. +-commutative 47.1%

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

      5. add-sqr-sqrt 47.1%

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

      6. hypot-define 47.1%

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

      7. associate-*l* 47.1%

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

      8. sqrt-prod 47.1%

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

      9. metadata-eval 47.1%

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

      10. sqrt-unprod 47.1%

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

      11. add-sqr-sqrt 47.1%

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

      12. metadata-eval 47.1%

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

   4. Applied egg-rr 47.1%

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

   5. Taylor expanded in x around -inf 57.3%

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

   if 5.39999999999999997e-6 < p < 3.3000000000000001e37

   1. Initial program 63.8%

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

   3. Taylor expanded in x around 0 62.5%

      \[\leadsto \sqrt{\color{blue}{0.5 + 0.25 \cdot \frac{x}{p}}} \]

   if 3.3000000000000001e37 < p

   1. Initial program 93.7%

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

   3. Taylor expanded in x around 0 89.8%

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

Alternative 5: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 7.2 \cdot 10^{-138}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 1.3 \cdot 10^{-51}:\\ \;\;\;\;-1 \cdot \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 7.2e-138)
   1.0
   (if (<= p_m 1.3e-51) (* -1.0 (/ p_m x)) (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 7.2e-138) {
		tmp = 1.0;
	} else if (p_m <= 1.3e-51) {
		tmp = -1.0 * (p_m / x);
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 7.2d-138) then
        tmp = 1.0d0
    else if (p_m <= 1.3d-51) then
        tmp = (-1.0d0) * (p_m / x)
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 7.2e-138) {
		tmp = 1.0;
	} else if (p_m <= 1.3e-51) {
		tmp = -1.0 * (p_m / x);
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 7.2e-138:
		tmp = 1.0
	elif p_m <= 1.3e-51:
		tmp = -1.0 * (p_m / x)
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 7.2e-138)
		tmp = 1.0;
	elseif (p_m <= 1.3e-51)
		tmp = Float64(-1.0 * Float64(p_m / x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 7.2e-138)
		tmp = 1.0;
	elseif (p_m <= 1.3e-51)
		tmp = -1.0 * (p_m / x);
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 7.2e-138], 1.0, If[LessEqual[p$95$m, 1.3e-51], N[(-1.0 * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 7.20000000000000036e-138

   1. Initial program 80.8%

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

      1. +-commutative 80.8%

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

      2. distribute-lft-in 80.8%

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

      3. associate-*r/ 80.8%

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

      4. +-commutative 80.8%

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

      5. add-sqr-sqrt 80.8%

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

      6. hypot-define 80.8%

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

      7. associate-*l* 80.8%

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

      8. sqrt-prod 80.8%

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

      9. metadata-eval 80.8%

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

      10. sqrt-unprod 12.9%

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

      11. add-sqr-sqrt 80.8%

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

      12. metadata-eval 80.8%

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

   4. Applied egg-rr 80.8%

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

   5. Taylor expanded in x around inf 40.1%

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

   if 7.20000000000000036e-138 < p < 1.3e-51

   1. Initial program 48.4%

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

      1. +-commutative 48.4%

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

      2. distribute-lft-in 48.4%

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

      3. associate-*r/ 48.4%

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

      4. +-commutative 48.4%

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

      5. add-sqr-sqrt 48.4%

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

      6. hypot-define 48.4%

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

      7. associate-*l* 48.4%

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

      8. sqrt-prod 48.4%

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

      9. metadata-eval 48.4%

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

      10. sqrt-unprod 48.4%

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

      11. add-sqr-sqrt 48.4%

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

      12. metadata-eval 48.4%

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

   4. Applied egg-rr 48.4%

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

   5. Taylor expanded in x around -inf 55.7%

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

   if 1.3e-51 < p

   1. Initial program 90.7%

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

   3. Taylor expanded in x around 0 80.2%

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

Alternative 6: 56.2% accurate, 21.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-233}:\\ \;\;\;\;-1 \cdot \frac{p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -5.2e-233) (* -1.0 (/ p_m x)) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -5.2e-233) {
		tmp = -1.0 * (p_m / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.2d-233)) then
        tmp = (-1.0d0) * (p_m / x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -5.2e-233) {
		tmp = -1.0 * (p_m / x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -5.2e-233:
		tmp = -1.0 * (p_m / x)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -5.2e-233)
		tmp = Float64(-1.0 * Float64(p_m / x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -5.2e-233)
		tmp = -1.0 * (p_m / x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -5.2e-233], N[(-1.0 * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.1999999999999996e-233

   1. Initial program 59.3%

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

      1. +-commutative 59.3%

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

      2. distribute-lft-in 59.3%

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

      3. associate-*r/ 59.3%

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

      4. +-commutative 59.3%

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

      5. add-sqr-sqrt 59.3%

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

      6. hypot-define 59.3%

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

      7. associate-*l* 59.3%

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

      8. sqrt-prod 59.3%

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

      9. metadata-eval 59.3%

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

      10. sqrt-unprod 30.7%

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

      11. add-sqr-sqrt 59.3%

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

      12. metadata-eval 59.3%

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

   4. Applied egg-rr 59.3%

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

   5. Taylor expanded in x around -inf 32.5%

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

   if -5.1999999999999996e-233 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. associate-*r/ 100.0%

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

      4. +-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. hypot-define 100.0%

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

      7. associate-*l* 100.0%

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

      8. sqrt-prod 100.0%

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

      9. metadata-eval 100.0%

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

      10. sqrt-unprod 40.9%

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

      11. add-sqr-sqrt 100.0%

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

      12. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 58.8%

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

Alternative 7: 36.4% accurate, 215.0× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x) :precision binary64 1.0)

C

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

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

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

Python

p_m = math.fabs(p)
def code(p_m, x):
	return 1.0

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := 1.0

Derivation

1. Initial program 80.3%

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

   1. +-commutative 80.3%

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

   2. distribute-lft-in 80.3%

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

   3. associate-*r/ 80.3%

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

   4. +-commutative 80.3%

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

   5. add-sqr-sqrt 80.3%

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

   6. hypot-define 80.3%

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

   7. associate-*l* 80.3%

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

   8. sqrt-prod 80.3%

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

   9. metadata-eval 80.3%

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

   10. sqrt-unprod 36.0%

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

   11. add-sqr-sqrt 80.3%

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

   12. metadata-eval 80.3%

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

4. Applied egg-rr 80.3%

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

5. Taylor expanded in x around inf 36.4%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer target: 79.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024052 -o generate:simplify
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (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))))))))