Given's Rotation SVD example

Percentage Accurate: 79.7% → 99.7%

Time: 11.0s

Alternatives: 10

Speedup: 0.6×

• could not determine a ground truth

  1. p = 1.698543244744012e-163
  2. x = -4.061364607926435e+108

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: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -1.0)
   (/ (- p_m) x)
   (sqrt
    (fma (/ x (/ 1.0 (sqrt (/ 1.0 (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(((p_m * (4.0 * p_m)) + (x * x)))) <= -1.0) {
		tmp = -p_m / x;
	} else {
		tmp = sqrt(fma((x / (1.0 / sqrt((1.0 / 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(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -1.0)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / Float64(1.0 / sqrt(Float64(1.0 / fma(p_m, Float64(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[(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[(N[(x / N[(1.0 / N[Sqrt[N[(1.0 / N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $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)))) < -1

   1. Initial program 14.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 43.2

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

   5. Simplified 43.2%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 54.7

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

   8. Simplified 54.7%

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

   if -1 < (/.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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. clear-num N/A

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

   6. Applied egg-rr 99.9%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{x}{\color{blue}{\frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(p, p \cdot 4, x \cdot x\right)}}}}}, 0.5, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.9%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0004:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(0.5, p\_m, x \cdot 0.25\right)}{p\_m}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p\_m \cdot p\_m}{x \cdot x}, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.5)
     (/ (- p_m) x)
     (if (<= t_0 0.0004)
       (sqrt (/ (fma 0.5 p_m (* x 0.25)) p_m))
       (fma -0.5 (/ (* p_m p_m) (* x x)) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.0004) {
		tmp = sqrt((fma(0.5, p_m, (x * 0.25)) / p_m));
	} else {
		tmp = fma(-0.5, ((p_m * p_m) / (x * x)), 1.0);
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.0004)
		tmp = sqrt(Float64(fma(0.5, p_m, Float64(x * 0.25)) / p_m));
	else
		tmp = fma(-0.5, Float64(Float64(p_m * p_m) / Float64(x * 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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0004], N[Sqrt[N[(N[(0.5 * p$95$m + N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / p$95$m), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $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 17.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.3

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

   8. Simplified 53.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000019e-4

   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 x around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in p around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 97.8

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

   11. Simplified 97.8%

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

   if 4.00000000000000019e-4 < (/.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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.3

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

   7. Simplified 99.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.0004:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(0.5, p, x \cdot 0.25\right)}{p}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p \cdot p}{x \cdot x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0004:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p\_m \cdot p\_m}{x \cdot x}, 1\right)\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.5)
     (/ (- p_m) x)
     (if (<= t_0 0.0004)
       (sqrt (fma 0.25 (/ x p_m) 0.5))
       (fma -0.5 (/ (* p_m p_m) (* x x)) 1.0)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.0004) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = fma(-0.5, ((p_m * p_m) / (x * x)), 1.0);
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.0004)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = fma(-0.5, Float64(Float64(p_m * p_m) / Float64(x * 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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0004], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(-0.5 * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $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 17.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.3

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

   8. Simplified 53.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000019e-4

   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 x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 97.8

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

   5. Simplified 97.8%

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

   if 4.00000000000000019e-4 < (/.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-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.3

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

   7. Simplified 99.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.0004:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{p \cdot p}{x \cdot x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0004:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.5)
     (/ (- p_m) x)
     (if (<= t_0 0.0004) (sqrt (fma 0.25 (/ x p_m) 0.5)) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.0004) {
		tmp = sqrt(fma(0.25, (x / p_m), 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(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.0004)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0004], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 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 17.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.3

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

   8. Simplified 53.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000019e-4

   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 x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 97.8

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

   5. Simplified 97.8%

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

   if 4.00000000000000019e-4 < (/.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 x around inf

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

   5. Simplified 98.8%

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

      1. metadata-eval 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.0004:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;\frac{-p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0004:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -0.5) (/ (- p_m) x) (if (<= t_0 0.0004) (sqrt 0.5) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.0004) {
		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(((p_m * (4.0d0 * p_m)) + (x * x)))
    if (t_0 <= (-0.5d0)) then
        tmp = -p_m / x
    else if (t_0 <= 0.0004d0) 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(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -p_m / x;
	} else if (t_0 <= 0.0004) {
		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(((p_m * (4.0 * p_m)) + (x * x)))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -p_m / x
	elif t_0 <= 0.0004:
		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(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	elseif (t_0 <= 0.0004)
		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(((p_m * (4.0 * p_m)) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -p_m / x;
	elseif (t_0 <= 0.0004)
		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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[((-p$95$m) / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0004], 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 17.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.3

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

   8. Simplified 53.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000019e-4

   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 x around 0

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

   5. Simplified 97.0%

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

   if 4.00000000000000019e-4 < (/.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 x around inf

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

   5. Simplified 98.8%

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

      1. metadata-eval 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;\frac{-p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.0004:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 0.0004:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x))))))
   (if (<= t_0 -1.0) (/ p_m x) (if (<= t_0 0.0004) (sqrt 0.5) 1.0))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = x / sqrt(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -1.0) {
		tmp = p_m / x;
	} else if (t_0 <= 0.0004) {
		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(((p_m * (4.0d0 * p_m)) + (x * x)))
    if (t_0 <= (-1.0d0)) then
        tmp = p_m / x
    else if (t_0 <= 0.0004d0) 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(((p_m * (4.0 * p_m)) + (x * x)));
	double tmp;
	if (t_0 <= -1.0) {
		tmp = p_m / x;
	} else if (t_0 <= 0.0004) {
		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(((p_m * (4.0 * p_m)) + (x * x)))
	tmp = 0
	if t_0 <= -1.0:
		tmp = p_m / x
	elif t_0 <= 0.0004:
		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(p_m * Float64(4.0 * p_m)) + Float64(x * x))))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(p_m / x);
	elseif (t_0 <= 0.0004)
		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(((p_m * (4.0 * p_m)) + (x * x)));
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = p_m / x;
	elseif (t_0 <= 0.0004)
		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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[(p$95$m / x), $MachinePrecision], If[LessEqual[t$95$0, 0.0004], 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)))) < -1

   1. Initial program 14.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 43.2

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

   5. Simplified 43.2%

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

      1. times-frac N/A

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

      2. sqrt-prod N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-/.f64 58.1

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

   7. Applied egg-rr 58.1%

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

   if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 4.00000000000000019e-4

   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. Taylor expanded in x around 0

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

   5. Simplified 95.7%

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

   if 4.00000000000000019e-4 < (/.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 x around inf

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

   5. Simplified 98.8%

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

      1. metadata-eval 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -1:\\ \;\;\;\;\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 0.0004:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / N[Sqrt[N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(x * x + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x + 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)))) < -1

   1. Initial program 14.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 43.2

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

   5. Simplified 43.2%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 54.7

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

   8. Simplified 54.7%

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

   if -1 < (/.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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 90.9%

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

Alternative 8: 98.5% accurate, 0.6× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -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(((p_m * (4.0 * p_m)) + (x * x)))) <= -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(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.5)
		tmp = Float64(Float64(-p_m) / x);
	else
		tmp = sqrt(fma(Float64(x / fma(p_m, Float64(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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], N[((-p$95$m) / x), $MachinePrecision], N[Sqrt[N[(N[(x / N[(p$95$m * N[(p$95$m * N[(2.0 / x), $MachinePrecision]), $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 17.3%

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

   3. Taylor expanded in x around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{\color{blue}{p \cdot p}}{{x}^{2}}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 42.6

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

   5. Simplified 42.6%

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

   6. Taylor expanded in p around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 53.3

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

   8. Simplified 53.3%

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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 97.9

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

   7. Simplified 97.9%

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

4. Final simplification 88.6%

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

Alternative 9: 75.3% accurate, 1.0× speedup

Math

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

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) 0.46) (sqrt 0.5) 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)))) <= 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(((p_m * (4.0d0 * p_m)) + (x * x)))) <= 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(((p_m * (4.0 * p_m)) + (x * x)))) <= 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(((p_m * (4.0 * p_m)) + (x * x)))) <= 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(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= 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(((p_m * (4.0 * p_m)) + (x * x)))) <= 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[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $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 75.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 70.0%

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

   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. Taylor expanded in x around inf

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

   5. Simplified 98.8%

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

      1. metadata-eval 98.8

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

   7. Applied egg-rr 98.8%

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

4. Final simplification 78.5%

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

Alternative 10: 36.6% 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 82.9%

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

3. Taylor expanded in x around inf

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

5. Simplified 40.3%

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

   1. metadata-eval 40.3

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

7. Applied egg-rr 40.3%

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

Developer Target 1: 79.7% 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 2024207 
(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))))))))