Given's Rotation SVD example

Percentage Accurate: 78.8% → 99.9%

Time: 9.8s

Alternatives: 9

Speedup: 0.6×

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: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}} \leq -0.999996:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot 4, p\_m, x \cdot x\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m)))) -0.999996)
   (/ (- p_m) x)
   (exp
    (* (log (fma (/ x (sqrt (fma (* p_m 4.0) p_m (* x x)))) 0.5 0.5)) 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= -0.999996) {
		tmp = -p_m / x;
	} else {
		tmp = exp((log(fma((x / sqrt(fma((p_m * 4.0), p_m, (x * x)))), 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(x * x) + Float64(Float64(p_m * 4.0) * p_m)))) <= -0.999996)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = exp(Float64(log(fma(Float64(x / sqrt(fma(Float64(p_m * 4.0), p_m, Float64(x * x)))), 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[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.999996], N[((-p$95$m) / x), $MachinePrecision], N[Exp[N[(N[Log[N[(N[(x / N[Sqrt[N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]], $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 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.999995999999999996

   1. Initial program 20.5%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 20.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.999995999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 99.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq -0.999996:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)}}, 0.5, 0.5\right)\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{x}, \frac{p\_m \cdot p\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m))))))
   (if (<= t_0 -0.5)
     (/ (- p_m) x)
     (if (<= t_0 2e-18)
       (sqrt (fma (/ x p_m) 0.25 0.5))
       (fma (/ -0.5 x) (/ (* p_m p_m) x) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((x * x) + ((p_m * 4.0) * p_m)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 2e-18) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = fma((-0.5 / x), ((p_m * p_m) / x), 1.0);
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(x * x) + Float64(Float64(p_m * 4.0) * p_m))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 2e-18)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = fma(Float64(-0.5 / x), Float64(Float64(p_m * p_m) / x), 1.0);
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-18], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(-0.5 / x), $MachinePrecision] * N[(N[(p$95$m * p$95$m), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 21.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 21.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 2.0000000000000001e-18

   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. Taylor expanded in p around inf

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{x}{p} + \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{p} \cdot \frac{1}{4}} + \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, \frac{1}{4}, \frac{1}{2}\right)}} \]

      4. lower-/.f64 99.8

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{p}}, 0.25, 0.5\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]

   if 2.0000000000000001e-18 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.5}{x}, \frac{p \cdot p}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(p\_m \cdot p\_m\right) \cdot -0.5\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m))))))
   (if (<= t_0 -0.5)
     (/ (- p_m) x)
     (if (<= t_0 2e-18)
       (sqrt (fma (/ x p_m) 0.25 0.5))
       (/ (fma x x (* (* p_m p_m) -0.5)) (* x x))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((x * x) + ((p_m * 4.0) * p_m)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 2e-18) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = fma(x, x, ((p_m * p_m) * -0.5)) / (x * x);
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(x * x) + Float64(Float64(p_m * 4.0) * p_m))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 2e-18)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = Float64(fma(x, x, Float64(Float64(p_m * p_m) * -0.5)) / Float64(x * x));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-18], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(x * x + N[(N[(p$95$m * p$95$m), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 21.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 21.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 2.0000000000000001e-18

   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. Taylor expanded in p around inf

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{x}{p} + \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{p} \cdot \frac{1}{4}} + \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, \frac{1}{4}, \frac{1}{2}\right)}} \]

      4. lower-/.f64 99.8

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{p}}, 0.25, 0.5\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]

   if 2.0000000000000001e-18 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(p \cdot p\right) \cdot -0.5\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p\_m}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m))))))
   (if (<= t_0 -0.5)
     (/ (- p_m) x)
     (if (<= t_0 2e-18) (sqrt (fma (/ x p_m) 0.25 0.5)) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((x * x) + ((p_m * 4.0) * p_m)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 2e-18) {
		tmp = sqrt(fma((x / p_m), 0.25, 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(x * x) + Float64(Float64(p_m * 4.0) * p_m))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 2e-18)
		tmp = sqrt(fma(Float64(x / p_m), 0.25, 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-18], N[Sqrt[N[(N[(x / p$95$m), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 21.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 21.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 2.0000000000000001e-18

   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. Taylor expanded in p around inf

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{x}{p} + \frac{1}{2}}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{p} \cdot \frac{1}{4}} + \frac{1}{2}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, \frac{1}{4}, \frac{1}{2}\right)}} \]

      4. lower-/.f64 99.8

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{p}}, 0.25, 0.5\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}} \]

   if 2.0000000000000001e-18 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in p around 0

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

   7. Applied rewrites 98.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{p}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m))))))
   (if (<= t_0 -0.5) (/ (- p_m) x) (if (<= t_0 2e-18) (sqrt 0.5) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((x * x) + ((p_m * 4.0) * p_m)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 2e-18) {
		tmp = sqrt(0.5);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x / sqrt(((x * x) + ((p_m * 4.0d0) * p_m)))
    if (t_0 <= (-0.5d0)) then
        tmp = -p_m / x
    else if (t_0 <= 2d-18) then
        tmp = sqrt(0.5d0)
    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 t_0 = x / Math.sqrt(((x * x) + ((p_m * 4.0) * p_m)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 2e-18) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = x / math.sqrt(((x * x) + ((p_m * 4.0) * p_m)))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -p_m / x
	elif t_0 <= 2e-18:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(x * x) + Float64(Float64(p_m * 4.0) * p_m))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 2e-18)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = x / sqrt(((x * x) + ((p_m * 4.0) * p_m)));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -p_m / x;
	elseif (t_0 <= 2e-18)
		tmp = sqrt(0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(x / N[Sqrt[N[(N[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 2e-18], N[Sqrt[0.5], $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 21.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 21.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 2.0000000000000001e-18

   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. Taylor expanded in p around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.5%

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

   if 2.0000000000000001e-18 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in p around 0

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

   7. Applied rewrites 98.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}} \leq -0.999996:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p\_m \cdot 4, p\_m, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m)))) -0.999996)
   (/ (- p_m) x)
   (sqrt (fma (/ x (sqrt (fma (* p_m 4.0) p_m (* x x)))) 0.5 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= -0.999996) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / sqrt(fma((p_m * 4.0), p_m, (x * x)))), 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(x * x) + Float64(Float64(p_m * 4.0) * p_m)))) <= -0.999996)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(Float64(p_m * 4.0), p_m, Float64(x * x)))), 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[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.999996], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[Sqrt[N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.999995999999999996

   1. Initial program 20.5%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 20.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.999995999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} + 1\right)}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      6. lower-fma.f64 99.9

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 99.9

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

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(\color{blue}{4 \cdot p}, p, x \cdot x\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      11. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(\color{blue}{p \cdot 4}, p, x \cdot x\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]

      12. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq -0.999996:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{x}{\sqrt{\mathsf{fma}\left(p \cdot 4, p, x \cdot x\right)}}, 0.5, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= -0.5) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / fma((p_m * p_m), (2.0 / x), x)), 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(x * x) + Float64(Float64(p_m * 4.0) * p_m)))) <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / fma(Float64(p_m * p_m), Float64(2.0 / x), x)), 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[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[(N[(p$95$m * p$95$m), $MachinePrecision] * N[(2.0 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < -0.5

   1. Initial program 21.3%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 21.3%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(p\right)}}{x} \]

      4. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

   3. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.3

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

   7. Applied rewrites 99.3%

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

4. Final simplification 84.5%

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

Alternative 8: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p\_m \cdot 4\right) \cdot p\_m}} \leq 0.46:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* x x) (* (* p_m 4.0) p_m)))) 0.46) (sqrt 0.5) 1.0))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= 0.46) {
		tmp = sqrt(0.5);
	} 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 / sqrt(((x * x) + ((p_m * 4.0d0) * p_m)))) <= 0.46d0) then
        tmp = sqrt(0.5d0)
    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 / Math.sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= 0.46) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= 0.46:
		tmp = math.sqrt(0.5)
	else:
		tmp = 1.0
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(x * x) + Float64(Float64(p_m * 4.0) * p_m)))) <= 0.46)
		tmp = sqrt(0.5);
	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 / sqrt(((x * x) + ((p_m * 4.0) * p_m)))) <= 0.46)
		tmp = sqrt(0.5);
	else
		tmp = 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[(x * x), $MachinePrecision] + N[(N[(p$95$m * 4.0), $MachinePrecision] * p$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.46], N[Sqrt[0.5], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 0.46000000000000002

   1. Initial program 70.7%

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

   3. Taylor expanded in p around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.6%

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

   if 0.46000000000000002 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x))))

   1. Initial program 100.0%

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

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

      2. pow1/2 N/A

         \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      4. lower-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

      5. lower-*.f64 N/A

         \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in p around 0

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

   7. Applied rewrites 98.7%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{x \cdot x + \left(p \cdot 4\right) \cdot p}} \leq 0.46:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.9% accurate, 58.0× 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 78.2%

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

   1. lift-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}} \]

   2. pow1/2 N/A

      \[\leadsto \color{blue}{{\left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right)}^{\frac{1}{2}}} \]

   3. pow-to-exp N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   4. lower-exp.f64 N/A

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

   5. lower-*.f64 N/A

      \[\leadsto e^{\color{blue}{\log \left(\frac{1}{2} \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)\right) \cdot \frac{1}{2}}} \]

4. Applied rewrites 78.2%

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

5. Taylor expanded in p around 0

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

7. Applied rewrites 36.0%

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

Developer Target 1: 78.8% accurate, 0.2× 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 2024277 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (! :herbie-platform default (sqrt (+ 1/2 (/ (copysign 1/2 x) (hypot 1 (/ (* 2 p) x))))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))