Given's Rotation SVD example

Percentage Accurate: 79.6% → 99.8%

Time: 12.5s

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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(log(exp(fma((x / hypot(x, (p_m * 2.0))), 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(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(log(exp(fma(Float64(x / hypot(x, Float64(p_m * 2.0))), 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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[Log[N[Exp[N[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $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 7.8%

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

      1. +-commutative 7.8%

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

      2. sqr-neg 7.8%

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

      3. associate-*l* 7.8%

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

      4. sqr-neg 7.8%

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

      5. fma-def 7.8%

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

      6. sqr-neg 7.8%

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

      7. fma-def 7.8%

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

      8. associate-*l* 7.8%

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

      9. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in x around -inf 52.2%

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

      1. associate-*r/ 52.2%

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

      2. mul-1-neg 52.2%

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

   7. Simplified 52.2%

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

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

   1. Initial program 99.8%

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

      1. +-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. sqr-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. sqr-neg 99.8%

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

      5. fma-def 99.8%

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

      6. sqr-neg 99.8%

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

      7. fma-def 99.8%

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

      8. associate-*l* 99.8%

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

      9. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. fma-udef 99.8%

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

      5. associate-*r* 99.8%

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

      6. +-commutative 99.8%

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

      7. fma-def 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 88.5%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p_m \cdot \left(4 \cdot p_m\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p_m \cdot 2\right)}\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (sqrt (* 0.5 (exp (log1p (/ x (hypot x (* p_m 2.0)))))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt((0.5 * exp(log1p((x / hypot(x, (p_m * 2.0)))))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = Math.sqrt((0.5 * Math.exp(Math.log1p((x / Math.hypot(x, (p_m * 2.0)))))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0:
		tmp = -p_m / x
	else:
		tmp = math.sqrt((0.5 * math.exp(math.log1p((x / math.hypot(x, (p_m * 2.0)))))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(Float64(0.5 * exp(log1p(Float64(x / hypot(x, Float64(p_m * 2.0)))))));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[Exp[N[Log[1 + N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $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 7.8%

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

      1. +-commutative 7.8%

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

      2. sqr-neg 7.8%

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

      3. associate-*l* 7.8%

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

      4. sqr-neg 7.8%

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

      5. fma-def 7.8%

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

      6. sqr-neg 7.8%

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

      7. fma-def 7.8%

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

      8. associate-*l* 7.8%

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

      9. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in x around -inf 52.2%

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

      1. associate-*r/ 52.2%

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

      2. mul-1-neg 52.2%

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

   7. Simplified 52.2%

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

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

   1. Initial program 99.8%

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

      1. flip3-+ 99.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}} \]

      2. add-exp-log 99.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\log \left(\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} - 1 \cdot \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}}} \]

      3. flip3-+ 99.8%

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

      4. log1p-udef 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*r* 99.8%

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

      7. fma-udef 99.8%

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

   4. Applied egg-rr 99.8%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\mathsf{log1p}\left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0)
		tmp = -p_m / x;
	else
		tmp = sqrt((0.5 * ((x / hypot(x, (p_m * 2.0))) + 1.0)));
	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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(0.5 * N[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 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)))) < -1

   1. Initial program 7.8%

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

      1. +-commutative 7.8%

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

      2. sqr-neg 7.8%

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

      3. associate-*l* 7.8%

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

      4. sqr-neg 7.8%

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

      5. fma-def 7.8%

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

      6. sqr-neg 7.8%

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

      7. fma-def 7.8%

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

      8. associate-*l* 7.8%

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

      9. +-commutative 7.8%

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

   3. Simplified 7.8%

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

   5. Taylor expanded in x around -inf 52.2%

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

      1. associate-*r/ 52.2%

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

      2. mul-1-neg 52.2%

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

   7. Simplified 52.2%

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

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

   1. Initial program 99.8%

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

      1. +-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. sqr-neg 99.8%

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

      3. associate-*l* 99.8%

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

      4. sqr-neg 99.8%

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

      5. fma-def 99.8%

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

      6. sqr-neg 99.8%

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

      7. fma-def 99.8%

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

      8. associate-*l* 99.8%

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

      9. +-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. fma-udef 99.8%

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

      3. associate-*r* 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-rgt1-in 99.8%

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

      6. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 88.5%

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

Alternative 4: 69.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ t_1 := 1 + \left(\frac{p_m}{x} \cdot \frac{p_m}{x}\right) \cdot -0.5\\ \mathbf{if}\;p_m \leq 3 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p_m \leq 1.4 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 2.4 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;p_m \leq 3.05 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)) (t_1 (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))))
   (if (<= p_m 3e-215)
     t_1
     (if (<= p_m 1.4e-170)
       t_0
       (if (<= p_m 2.4e-108) t_1 (if (<= p_m 3.05e-65) t_0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double t_1 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	double tmp;
	if (p_m <= 3e-215) {
		tmp = t_1;
	} else if (p_m <= 1.4e-170) {
		tmp = t_0;
	} else if (p_m <= 2.4e-108) {
		tmp = t_1;
	} else if (p_m <= 3.05e-65) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -p_m / x
    t_1 = 1.0d0 + (((p_m / x) * (p_m / x)) * (-0.5d0))
    if (p_m <= 3d-215) then
        tmp = t_1
    else if (p_m <= 1.4d-170) then
        tmp = t_0
    else if (p_m <= 2.4d-108) then
        tmp = t_1
    else if (p_m <= 3.05d-65) then
        tmp = t_0
    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 t_0 = -p_m / x;
	double t_1 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	double tmp;
	if (p_m <= 3e-215) {
		tmp = t_1;
	} else if (p_m <= 1.4e-170) {
		tmp = t_0;
	} else if (p_m <= 2.4e-108) {
		tmp = t_1;
	} else if (p_m <= 3.05e-65) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	t_1 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	tmp = 0
	if p_m <= 3e-215:
		tmp = t_1
	elif p_m <= 1.4e-170:
		tmp = t_0
	elif p_m <= 2.4e-108:
		tmp = t_1
	elif p_m <= 3.05e-65:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	t_1 = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5))
	tmp = 0.0
	if (p_m <= 3e-215)
		tmp = t_1;
	elseif (p_m <= 1.4e-170)
		tmp = t_0;
	elseif (p_m <= 2.4e-108)
		tmp = t_1;
	elseif (p_m <= 3.05e-65)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	t_1 = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	tmp = 0.0;
	if (p_m <= 3e-215)
		tmp = t_1;
	elseif (p_m <= 1.4e-170)
		tmp = t_0;
	elseif (p_m <= 2.4e-108)
		tmp = t_1;
	elseif (p_m <= 3.05e-65)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[p$95$m, 3e-215], t$95$1, If[LessEqual[p$95$m, 1.4e-170], t$95$0, If[LessEqual[p$95$m, 2.4e-108], t$95$1, If[LessEqual[p$95$m, 3.05e-65], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 3.00000000000000025e-215 or 1.39999999999999998e-170 < p < 2.40000000000000017e-108

   1. Initial program 77.6%

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

      1. +-commutative 77.6%

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

      2. sqr-neg 77.6%

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

      3. associate-*l* 77.6%

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

      4. sqr-neg 77.6%

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

      5. fma-def 77.6%

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

      6. sqr-neg 77.6%

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

      7. fma-def 77.6%

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

      8. associate-*l* 77.6%

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

      9. +-commutative 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in x around inf 33.7%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 33.7%

         \[\leadsto 1 + \color{blue}{\frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]

   7. Simplified 33.7%

      \[\leadsto \color{blue}{1 + \frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]
   8. Step-by-step derivation

      1. unpow2 33.7%

         \[\leadsto 1 + \frac{\color{blue}{p \cdot p}}{{x}^{2}} \cdot -0.5 \]

      2. unpow2 33.7%

         \[\leadsto 1 + \frac{p \cdot p}{\color{blue}{x \cdot x}} \cdot -0.5 \]

      3. times-frac 33.7%

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

   9. Applied egg-rr 33.7%

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

   if 3.00000000000000025e-215 < p < 1.39999999999999998e-170 or 2.40000000000000017e-108 < p < 3.05000000000000007e-65

   1. Initial program 40.5%

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

      1. +-commutative 40.5%

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

      2. sqr-neg 40.5%

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

      3. associate-*l* 40.5%

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

      4. sqr-neg 40.5%

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

      5. fma-def 40.5%

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

      6. sqr-neg 40.5%

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

      7. fma-def 40.5%

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

      8. associate-*l* 40.5%

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

      9. +-commutative 40.5%

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

   3. Simplified 40.5%

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

   5. Taylor expanded in x around -inf 63.9%

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

      1. associate-*r/ 63.9%

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

      2. mul-1-neg 63.9%

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

   7. Simplified 63.9%

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

   if 3.05000000000000007e-65 < p

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0 87.7%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 3 \cdot 10^{-215}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p \leq 1.4 \cdot 10^{-170}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.4 \cdot 10^{-108}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p \leq 3.05 \cdot 10^{-65}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{-p_m}{x}\\ \mathbf{if}\;p_m \leq 4.8 \cdot 10^{-215}:\\ \;\;\;\;1 + \left(\frac{p_m}{x} \cdot \frac{p_m}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p_m \leq 1.35 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;p_m \leq 2.8 \cdot 10^{-106}:\\ \;\;\;\;1\\ \mathbf{elif}\;p_m \leq 4.1 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ (- p_m) x)))
   (if (<= p_m 4.8e-215)
     (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))
     (if (<= p_m 1.35e-170)
       t_0
       (if (<= p_m 2.8e-106) 1.0 (if (<= p_m 4.1e-66) t_0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = -p_m / x;
	double tmp;
	if (p_m <= 4.8e-215) {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	} else if (p_m <= 1.35e-170) {
		tmp = t_0;
	} else if (p_m <= 2.8e-106) {
		tmp = 1.0;
	} else if (p_m <= 4.1e-66) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = -p_m / x
    if (p_m <= 4.8d-215) then
        tmp = 1.0d0 + (((p_m / x) * (p_m / x)) * (-0.5d0))
    else if (p_m <= 1.35d-170) then
        tmp = t_0
    else if (p_m <= 2.8d-106) then
        tmp = 1.0d0
    else if (p_m <= 4.1d-66) then
        tmp = t_0
    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 t_0 = -p_m / x;
	double tmp;
	if (p_m <= 4.8e-215) {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	} else if (p_m <= 1.35e-170) {
		tmp = t_0;
	} else if (p_m <= 2.8e-106) {
		tmp = 1.0;
	} else if (p_m <= 4.1e-66) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = -p_m / x
	tmp = 0
	if p_m <= 4.8e-215:
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	elif p_m <= 1.35e-170:
		tmp = t_0
	elif p_m <= 2.8e-106:
		tmp = 1.0
	elif p_m <= 4.1e-66:
		tmp = t_0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(Float64(-p_m) / x)
	tmp = 0.0
	if (p_m <= 4.8e-215)
		tmp = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5));
	elseif (p_m <= 1.35e-170)
		tmp = t_0;
	elseif (p_m <= 2.8e-106)
		tmp = 1.0;
	elseif (p_m <= 4.1e-66)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = -p_m / x;
	tmp = 0.0;
	if (p_m <= 4.8e-215)
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	elseif (p_m <= 1.35e-170)
		tmp = t_0;
	elseif (p_m <= 2.8e-106)
		tmp = 1.0;
	elseif (p_m <= 4.1e-66)
		tmp = t_0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[((-p$95$m) / x), $MachinePrecision]}, If[LessEqual[p$95$m, 4.8e-215], N[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[p$95$m, 1.35e-170], t$95$0, If[LessEqual[p$95$m, 2.8e-106], 1.0, If[LessEqual[p$95$m, 4.1e-66], t$95$0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if p < 4.8000000000000002e-215

   1. Initial program 78.0%

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

      1. +-commutative 78.0%

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

      2. sqr-neg 78.0%

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

      3. associate-*l* 78.0%

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

      4. sqr-neg 78.0%

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

      5. fma-def 78.0%

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

      6. sqr-neg 78.0%

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

      7. fma-def 78.0%

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

      8. associate-*l* 78.0%

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

      9. +-commutative 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in x around inf 31.2%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 31.2%

         \[\leadsto 1 + \color{blue}{\frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]

   7. Simplified 31.2%

      \[\leadsto \color{blue}{1 + \frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]
   8. Step-by-step derivation

      1. unpow2 31.2%

         \[\leadsto 1 + \frac{\color{blue}{p \cdot p}}{{x}^{2}} \cdot -0.5 \]

      2. unpow2 31.2%

         \[\leadsto 1 + \frac{p \cdot p}{\color{blue}{x \cdot x}} \cdot -0.5 \]

      3. times-frac 31.2%

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

   9. Applied egg-rr 31.2%

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

   if 4.8000000000000002e-215 < p < 1.3499999999999999e-170 or 2.79999999999999988e-106 < p < 4.09999999999999998e-66

   1. Initial program 37.9%

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

      1. +-commutative 37.9%

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

      2. sqr-neg 37.9%

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

      3. associate-*l* 37.9%

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

      4. sqr-neg 37.9%

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

      5. fma-def 37.9%

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

      6. sqr-neg 37.9%

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

      7. fma-def 37.9%

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

      8. associate-*l* 37.9%

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

      9. +-commutative 37.9%

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

   3. Simplified 37.9%

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

   5. Taylor expanded in x around -inf 66.3%

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

      1. associate-*r/ 66.3%

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

      2. mul-1-neg 66.3%

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

   7. Simplified 66.3%

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

   if 1.3499999999999999e-170 < p < 2.79999999999999988e-106

   1. Initial program 74.2%

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

   3. Taylor expanded in x around inf 66.8%

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

   if 4.09999999999999998e-66 < p

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0 87.7%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.8 \cdot 10^{-215}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \mathbf{elif}\;p \leq 1.35 \cdot 10^{-170}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;p \leq 2.8 \cdot 10^{-106}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 4.1 \cdot 10^{-66}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.0% accurate, 13.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-150}:\\ \;\;\;\;\frac{-p_m}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{p_m}{x} \cdot \frac{p_m}{x}\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x 3e-150) (/ (- p_m) x) (+ 1.0 (* (* (/ p_m x) (/ p_m x)) -0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= 3e-150) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -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 (x <= 3d-150) then
        tmp = -p_m / x
    else
        tmp = 1.0d0 + (((p_m / x) * (p_m / x)) * (-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 (x <= 3e-150) {
		tmp = -p_m / x;
	} else {
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= 3e-150:
		tmp = -p_m / x
	else:
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= 3e-150)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(p_m / x) * Float64(p_m / x)) * -0.5));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= 3e-150)
		tmp = -p_m / x;
	else
		tmp = 1.0 + (((p_m / x) * (p_m / x)) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, 3e-150], N[((-p$95$m) / x), $MachinePrecision], N[(1.0 + N[(N[(N[(p$95$m / x), $MachinePrecision] * N[(p$95$m / x), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0000000000000002e-150

   1. Initial program 61.5%

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

      1. +-commutative 61.5%

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

      2. sqr-neg 61.5%

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

      3. associate-*l* 61.5%

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

      4. sqr-neg 61.5%

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

      5. fma-def 61.5%

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

      6. sqr-neg 61.5%

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

      7. fma-def 61.5%

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

      8. associate-*l* 61.5%

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

      9. +-commutative 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around -inf 23.6%

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

      1. associate-*r/ 23.6%

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

      2. mul-1-neg 23.6%

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

   7. Simplified 23.6%

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

   if 3.0000000000000002e-150 < 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. +-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. sqr-neg 100.0%

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

      3. associate-*l* 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-def 100.0%

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

      6. sqr-neg 100.0%

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

      7. fma-def 100.0%

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

      8. associate-*l* 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 56.2%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{p}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 56.2%

         \[\leadsto 1 + \color{blue}{\frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]

   7. Simplified 56.2%

      \[\leadsto \color{blue}{1 + \frac{{p}^{2}}{{x}^{2}} \cdot -0.5} \]
   8. Step-by-step derivation

      1. unpow2 56.2%

         \[\leadsto 1 + \frac{\color{blue}{p \cdot p}}{{x}^{2}} \cdot -0.5 \]

      2. unpow2 56.2%

         \[\leadsto 1 + \frac{p \cdot p}{\color{blue}{x \cdot x}} \cdot -0.5 \]

      3. times-frac 56.2%

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

   9. Applied egg-rr 56.2%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-150}:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{p}{x} \cdot \frac{p}{x}\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.5% accurate, 53.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \frac{-p_m}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := N[((-p$95$m) / x), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. +-commutative 77.9%

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

   2. sqr-neg 77.9%

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

   3. associate-*l* 77.9%

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

   4. sqr-neg 77.9%

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

   5. fma-def 77.9%

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

   6. sqr-neg 77.9%

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

   7. fma-def 77.9%

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

   8. associate-*l* 77.9%

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

   9. +-commutative 77.9%

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

3. Simplified 77.9%

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

5. Taylor expanded in x around -inf 15.1%

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

   1. associate-*r/ 15.1%

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

   2. mul-1-neg 15.1%

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

7. Simplified 15.1%

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

8. Final simplification 15.1%

   \[\leadsto \frac{-p}{x} \]
9. Add Preprocessing

Developer target: 79.7% 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 2024019 
(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))))))))