Given's Rotation SVD example

Percentage Accurate: 79.1% → 99.7%

Time: 10.9s

Alternatives: 8

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: 79.1% 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, 0.6× speedup

Math

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

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)))) -0.5)
     (- (/ p_m x))
     (sqrt (fma (/ x (sqrt (fma x x t_0))) 0.5 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)))) <= -0.5) {
		tmp = -(p_m / x);
	} else {
		tmp = sqrt(fma((x / sqrt(fma(x, x, t_0))), 0.5, 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)))) <= -0.5)
		tmp = Float64(-Float64(p_m / x));
	else
		tmp = sqrt(fma(Float64(x / sqrt(fma(x, x, t_0))), 0.5, 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], -0.5], (-N[(p$95$m / x), $MachinePrecision]), N[Sqrt[N[(N[(x / N[Sqrt[N[(x * x + t$95$0), $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)))) < -0.5

   1. Initial program 12.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \color{blue}{1} \cdot \frac{p}{\mathsf{neg}\left(x\right)} \]

      14. *-lft-identity N/A

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

      15. frac-2neg N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   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 rewrites 100.0%

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

4. Final simplification 87.5%

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

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\mathsf{fma}\left(0.5, \frac{p\_m \cdot p\_m}{x \cdot x}, 0.5\right)}}\\ \end{array} \end{array} \]

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 1e-7)
       (sqrt (fma 0.25 (/ x p_m) 0.5))
       (sqrt (/ 0.5 (fma 0.5 (/ (* p_m p_m) (* x x)) 0.5)))))))

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 <= 1e-7) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = sqrt((0.5 / fma(0.5, ((p_m * p_m) / (x * x)), 0.5)));
	}
	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 <= 1e-7)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = sqrt(Float64(0.5 / fma(0.5, Float64(Float64(p_m * p_m) / Float64(x * x)), 0.5)));
	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, 1e-7], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.5 / N[(0.5 * N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $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 12.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \color{blue}{1} \cdot \frac{p}{\mathsf{neg}\left(x\right)} \]

      14. *-lft-identity N/A

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

      15. frac-2neg N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 9.9999999999999995e-8

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 9.9999999999999995e-8 < (/.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. flip-+ N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{\mathsf{fma}\left(0.5, \frac{p \cdot p}{x \cdot x}, 0.5\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p\_m \cdot p\_m}{x \cdot x}, -0.5, 1\right)\\ \end{array} \end{array} \]

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 1e-7)
       (sqrt (fma 0.25 (/ x p_m) 0.5))
       (fma (/ (* p_m p_m) (* x x)) -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 <= 1e-7) {
		tmp = sqrt(fma(0.25, (x / p_m), 0.5));
	} else {
		tmp = fma(((p_m * p_m) / (x * x)), -0.5, 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 <= 1e-7)
		tmp = sqrt(fma(0.25, Float64(x / p_m), 0.5));
	else
		tmp = fma(Float64(Float64(p_m * p_m) / Float64(x * x)), -0.5, 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, 1e-7], N[Sqrt[N[(0.25 * N[(x / p$95$m), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[(N[(N[(p$95$m * p$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.5 + 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 12.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \color{blue}{1} \cdot \frac{p}{\mathsf{neg}\left(x\right)} \]

      14. *-lft-identity N/A

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

      15. frac-2neg N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 9.9999999999999995e-8

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 9.9999999999999995e-8 < (/.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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in p around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{p \cdot p}{x \cdot x}, -0.5, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p\_m}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 1e-7) (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 <= 1e-7) {
		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 <= 1e-7)
		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, 1e-7], 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 12.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \color{blue}{1} \cdot \frac{p}{\mathsf{neg}\left(x\right)} \]

      14. *-lft-identity N/A

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

      15. frac-2neg N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 9.9999999999999995e-8

   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 9.9999999999999995e-8 < (/.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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 99.3%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 10^{-7}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{x}{p}, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-\frac{p\_m}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

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 1e-7) (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 <= 1e-7) {
		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 <= 1d-7) 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 <= 1e-7) {
		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 <= 1e-7:
		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 <= 1e-7)
		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 <= 1e-7)
		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, 1e-7], 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 12.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \color{blue}{1} \cdot \frac{p}{\mathsf{neg}\left(x\right)} \]

      14. *-lft-identity N/A

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

      15. frac-2neg N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   if -0.5 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 #s(literal 4 binary64) p) p) (*.f64 x x)))) < 9.9999999999999995e-8

   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. Applied rewrites 97.3%

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

   if 9.9999999999999995e-8 < (/.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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 99.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.5:\\ \;\;\;\;-\frac{p}{x}\\ \mathbf{elif}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.6× speedup

Math

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

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 (* 2.0 (/ 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)))) <= -0.5) {
		tmp = -(p_m / x);
	} else {
		tmp = sqrt(fma((x / fma(p_m, (2.0 * (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)))) <= -0.5)
		tmp = Float64(-Float64(p_m / x));
	else
		tmp = sqrt(fma(Float64(x / fma(p_m, Float64(2.0 * Float64(p_m / 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[(2.0 * N[(p$95$m / 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 12.2%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 56.7

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

   5. Applied rewrites 56.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. sqrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

          \[\leadsto \color{blue}{1} \cdot \frac{p}{\mathsf{neg}\left(x\right)} \]

      14. *-lft-identity N/A

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

      15. frac-2neg N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

      18. lower-/.f64 N/A

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

      19. lower-neg.f64 57.3

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

   7. Applied rewrites 57.3%

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

   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. 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. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 98.1

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

   7. Applied rewrites 98.1%

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

4. Final simplification 86.2%

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

Alternative 7: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) 1e-7) (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)))) <= 1e-7) {
		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)))) <= 1d-7) 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)))) <= 1e-7) {
		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)))) <= 1e-7:
		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)))) <= 1e-7)
		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)))) <= 1e-7)
		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], 1e-7], 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)))) < 9.9999999999999995e-8

   1. Initial program 66.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.4%

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

   if 9.9999999999999995e-8 < (/.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

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in p around 0

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

   8. Applied rewrites 99.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq 10^{-7}:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.1% 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 74.3%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 24.2

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

5. Applied rewrites 24.2%

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

6. Taylor expanded in p around 0

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

8. Applied rewrites 33.6%

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

Developer Target 1: 79.1% 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 2024216 
(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))))))))