Given's Rotation SVD example

Percentage Accurate: 79.5% → 99.7%

Time: 9.5s

Alternatives: 7

Speedup: 1.0×

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: 79.5% 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

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

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 0.5) (/ 1.0 (hypot x (* p_m 2.0))) 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 * 0.5), (1.0 / hypot(x, (p_m * 2.0))), 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(p_m / Float64(-x));
	else
		tmp = sqrt(fma(Float64(x * 0.5), Float64(1.0 / hypot(x, Float64(p_m * 2.0))), 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 * 0.5), $MachinePrecision] * N[(1.0 / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 17.8%

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

      1. +-commutative 17.8%

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

      2. frac-2neg 17.8%

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

      3. div-inv 14.9%

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

      4. fma-define 3.7%

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

      5. +-commutative 3.7%

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

      6. add-sqr-sqrt 3.7%

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

      7. hypot-define 3.7%

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

      8. associate-*l* 3.7%

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

      9. sqrt-prod 3.7%

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

      10. metadata-eval 3.7%

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

      11. sqrt-unprod 2.8%

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

      12. add-sqr-sqrt 3.7%

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

   4. Applied egg-rr 3.7%

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

      1. fma-undefine 14.9%

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

      2. +-commutative 14.9%

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

      3. distribute-lft-neg-out 14.9%

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

      4. unsub-neg 14.9%

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

      5. distribute-frac-neg2 14.9%

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

      6. distribute-neg-frac 14.9%

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

      7. metadata-eval 14.9%

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

   6. Simplified 14.9%

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

   7. Taylor expanded in x around -inf 52.9%

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

      1. *-commutative 52.9%

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

      2. associate-*l/ 52.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in p around -inf 63.8%

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

       1. associate-*r/ 63.8%

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

       2. neg-mul-1 63.8%

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

   12. Simplified 63.8%

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

   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. +-commutative 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. div-inv 99.9%

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

      4. associate-*r* 99.9%

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

      5. metadata-eval 99.9%

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

      6. fma-define 99.9%

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

      7. +-commutative 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. hypot-define 99.9%

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

      10. associate-*l* 99.9%

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

      11. sqrt-prod 99.9%

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

      12. metadata-eval 99.9%

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

      13. sqrt-unprod 56.6%

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

      14. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 91.1%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 17.8%

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

      1. +-commutative 17.8%

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

      2. frac-2neg 17.8%

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

      3. div-inv 14.9%

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

      4. fma-define 3.7%

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

      5. +-commutative 3.7%

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

      6. add-sqr-sqrt 3.7%

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

      7. hypot-define 3.7%

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

      8. associate-*l* 3.7%

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

      9. sqrt-prod 3.7%

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

      10. metadata-eval 3.7%

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

      11. sqrt-unprod 2.8%

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

      12. add-sqr-sqrt 3.7%

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

   4. Applied egg-rr 3.7%

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

      1. fma-undefine 14.9%

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

      2. +-commutative 14.9%

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

      3. distribute-lft-neg-out 14.9%

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

      4. unsub-neg 14.9%

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

      5. distribute-frac-neg2 14.9%

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

      6. distribute-neg-frac 14.9%

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

      7. metadata-eval 14.9%

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

   6. Simplified 14.9%

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

   7. Taylor expanded in x around -inf 52.9%

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

      1. *-commutative 52.9%

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

      2. associate-*l/ 52.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in p around -inf 63.8%

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

       1. associate-*r/ 63.8%

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

       2. neg-mul-1 63.8%

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

   12. Simplified 63.8%

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

   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. +-commutative 99.9%

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

      2. frac-2neg 99.9%

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

      3. div-inv 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. add-sqr-sqrt 99.9%

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

      7. hypot-define 99.9%

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

      8. associate-*l* 99.9%

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

      9. sqrt-prod 99.9%

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

      10. metadata-eval 99.9%

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

      11. sqrt-unprod 56.6%

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

      12. add-sqr-sqrt 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. unsub-neg 99.9%

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

      5. distribute-frac-neg2 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 91.1%

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

Alternative 3: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -2.15e+52)
   (/ p_m (- x))
   (sqrt (+ 0.5 (/ (* x 0.5) (hypot x (* p_m 2.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2.15e+52) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (x <= -2.15e+52) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt((0.5 + ((x * 0.5) / Math.hypot(x, (p_m * 2.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2.15e+52:
		tmp = p_m / -x
	else:
		tmp = math.sqrt((0.5 + ((x * 0.5) / math.hypot(x, (p_m * 2.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2.15e+52)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 + Float64(Float64(x * 0.5) / hypot(x, Float64(p_m * 2.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -2.15e+52)
		tmp = p_m / -x;
	else
		tmp = sqrt((0.5 + ((x * 0.5) / hypot(x, (p_m * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2.15e+52], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 + N[(N[(x * 0.5), $MachinePrecision] / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.15e52

   1. Initial program 50.9%

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

      1. +-commutative 50.9%

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

      2. frac-2neg 50.9%

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

      3. div-inv 46.4%

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

      4. fma-define 32.1%

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

      5. +-commutative 32.1%

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

      6. add-sqr-sqrt 32.1%

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

      7. hypot-define 32.1%

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

      8. associate-*l* 32.1%

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

      9. sqrt-prod 32.1%

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

      10. metadata-eval 32.1%

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

      11. sqrt-unprod 24.5%

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

      12. add-sqr-sqrt 32.1%

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

   4. Applied egg-rr 32.1%

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

      1. fma-undefine 46.4%

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

      2. +-commutative 46.4%

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

      3. distribute-lft-neg-out 46.4%

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

      4. unsub-neg 46.4%

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

      5. distribute-frac-neg2 46.4%

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

      6. distribute-neg-frac 46.4%

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

      7. metadata-eval 46.4%

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

   6. Simplified 46.4%

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

   7. Taylor expanded in x around -inf 51.9%

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

      1. *-commutative 51.9%

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

      2. associate-*l/ 51.9%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 51.9%

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

   9. Simplified 51.9%

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

   10. Taylor expanded in p around -inf 49.3%

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

       1. associate-*r/ 49.3%

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

       2. neg-mul-1 49.3%

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

   12. Simplified 49.3%

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

   if -2.15e52 < x

   1. Initial program 85.9%

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

      1. +-commutative 85.9%

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

      2. distribute-lft-in 85.9%

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

      3. associate-*r/ 85.9%

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

      4. +-commutative 85.9%

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

      5. add-sqr-sqrt 85.9%

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

      6. hypot-define 85.9%

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

      7. associate-*l* 85.9%

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

      8. sqrt-prod 85.9%

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

      9. metadata-eval 85.9%

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

      10. sqrt-unprod 47.6%

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

      11. add-sqr-sqrt 85.9%

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

      12. metadata-eval 85.9%

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

   4. Applied egg-rr 85.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+52}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 + \frac{x \cdot 0.5}{\mathsf{hypot}\left(x, p \cdot 2\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 1.75 \cdot 10^{-177}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{elif}\;p\_m \leq 1.46 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 1.75e-177) (/ p_m (- x)) (if (<= p_m 1.46e-74) 1.0 (sqrt 0.5))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.75e-177) {
		tmp = p_m / -x;
	} else if (p_m <= 1.46e-74) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 1.75d-177) then
        tmp = p_m / -x
    else if (p_m <= 1.46d-74) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 1.75e-177) {
		tmp = p_m / -x;
	} else if (p_m <= 1.46e-74) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 1.75e-177:
		tmp = p_m / -x
	elif p_m <= 1.46e-74:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 1.75e-177)
		tmp = Float64(p_m / Float64(-x));
	elseif (p_m <= 1.46e-74)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 1.75e-177)
		tmp = p_m / -x;
	elseif (p_m <= 1.46e-74)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 1.75e-177], N[(p$95$m / (-x)), $MachinePrecision], If[LessEqual[p$95$m, 1.46e-74], 1.0, N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.7500000000000001e-177

   1. Initial program 77.0%

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

      1. +-commutative 77.0%

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

      2. frac-2neg 77.0%

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

      3. div-inv 75.6%

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

      4. fma-define 70.3%

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

      5. +-commutative 70.3%

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

      6. add-sqr-sqrt 70.3%

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

      7. hypot-define 70.3%

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

      8. associate-*l* 70.3%

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

      9. sqrt-prod 70.3%

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

      10. metadata-eval 70.3%

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

      11. sqrt-unprod 10.8%

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

      12. add-sqr-sqrt 70.3%

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

   4. Applied egg-rr 70.3%

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

      1. fma-undefine 75.6%

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

      2. +-commutative 75.6%

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

      3. distribute-lft-neg-out 75.6%

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

      4. unsub-neg 75.6%

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

      5. distribute-frac-neg2 75.6%

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

      6. distribute-neg-frac 75.6%

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

      7. metadata-eval 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in x around -inf 15.0%

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

      1. *-commutative 15.0%

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

      2. associate-*l/ 15.0%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 15.0%

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

   9. Simplified 15.0%

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

   10. Taylor expanded in p around -inf 17.1%

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

       1. associate-*r/ 17.1%

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

       2. neg-mul-1 17.1%

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

   12. Simplified 17.1%

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

   if 1.7500000000000001e-177 < p < 1.46e-74

   1. Initial program 66.6%

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

      1. +-commutative 66.6%

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

      2. distribute-lft-in 66.6%

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

      3. associate-*r/ 66.6%

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

      4. +-commutative 66.6%

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

      5. add-sqr-sqrt 66.6%

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

      6. hypot-define 66.5%

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

      7. associate-*l* 66.5%

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

      8. sqrt-prod 66.5%

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

      9. metadata-eval 66.5%

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

      10. sqrt-unprod 66.5%

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

      11. add-sqr-sqrt 66.5%

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

      12. metadata-eval 66.5%

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

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in x around inf 61.0%

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

   if 1.46e-74 < p

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0 80.5%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.75 \cdot 10^{-177}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.46 \cdot 10^{-74}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 8.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 8.8e-77) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 8.8e-77) {
		tmp = p_m / -x;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

p_m = abs(p)
real(8) function code(p_m, x)
    real(8), intent (in) :: p_m
    real(8), intent (in) :: x
    real(8) :: tmp
    if (p_m <= 8.8d-77) then
        tmp = p_m / -x
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if (p_m <= 8.8e-77) {
		tmp = p_m / -x;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if p_m <= 8.8e-77:
		tmp = p_m / -x
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (p_m <= 8.8e-77)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (p_m <= 8.8e-77)
		tmp = p_m / -x;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[p$95$m, 8.8e-77], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 8.80000000000000028e-77

   1. Initial program 75.4%

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

      1. +-commutative 75.4%

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

      2. frac-2neg 75.4%

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

      3. div-inv 74.2%

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

      4. fma-define 69.7%

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

      5. +-commutative 69.7%

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

      6. add-sqr-sqrt 69.7%

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

      7. hypot-define 69.7%

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

      8. associate-*l* 69.7%

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

      9. sqrt-prod 69.7%

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

      10. metadata-eval 69.7%

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

      11. sqrt-unprod 18.2%

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

      12. add-sqr-sqrt 69.7%

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

   4. Applied egg-rr 69.7%

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

      1. fma-undefine 74.2%

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

      2. +-commutative 74.2%

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

      3. distribute-lft-neg-out 74.2%

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

      4. unsub-neg 74.2%

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

      5. distribute-frac-neg2 74.2%

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

      6. distribute-neg-frac 74.2%

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

      7. metadata-eval 74.2%

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

   6. Simplified 74.2%

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

   7. Taylor expanded in x around -inf 16.8%

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

      1. *-commutative 16.8%

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

      2. associate-*l/ 16.8%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 16.8%

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

   9. Simplified 16.8%

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

   10. Taylor expanded in p around -inf 19.9%

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

       1. associate-*r/ 19.9%

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

       2. neg-mul-1 19.9%

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

   12. Simplified 19.9%

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

   if 8.80000000000000028e-77 < p

   1. Initial program 88.2%

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

   3. Taylor expanded in x around 0 79.8%

      \[\leadsto \color{blue}{\sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 8.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.5% accurate, 23.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= x -2e-310) (/ p_m (- x)) (/ p_m x)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	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 <= (-2d-310)) then
        tmp = p_m / -x
    else
        tmp = p_m / x
    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 <= -2e-310) {
		tmp = p_m / -x;
	} else {
		tmp = p_m / x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if x <= -2e-310:
		tmp = p_m / -x
	else:
		tmp = p_m / x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(p_m / Float64(-x));
	else
		tmp = Float64(p_m / x);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = p_m / -x;
	else
		tmp = p_m / x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[x, -2e-310], N[(p$95$m / (-x)), $MachinePrecision], N[(p$95$m / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

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

      1. +-commutative 58.7%

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

      2. frac-2neg 58.7%

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

      3. div-inv 57.3%

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

      4. fma-define 51.7%

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

      5. +-commutative 51.7%

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

      6. add-sqr-sqrt 51.7%

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

      7. hypot-define 51.7%

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

      8. associate-*l* 51.7%

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

      9. sqrt-prod 51.7%

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

      10. metadata-eval 51.7%

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

      11. sqrt-unprod 33.5%

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

      12. add-sqr-sqrt 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. fma-undefine 57.3%

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

      2. +-commutative 57.3%

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

      3. distribute-lft-neg-out 57.3%

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

      4. unsub-neg 57.3%

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

      5. distribute-frac-neg2 57.3%

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

      6. distribute-neg-frac 57.3%

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

      7. metadata-eval 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in x around -inf 29.2%

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

      1. *-commutative 29.2%

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

      2. associate-*l/ 29.2%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in p around -inf 34.2%

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

       1. associate-*r/ 34.2%

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

       2. neg-mul-1 34.2%

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

   12. Simplified 34.2%

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

   if -1.999999999999994e-310 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. frac-2neg 100.0%

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

      3. div-inv 100.0%

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

      4. fma-define 100.0%

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

      5. +-commutative 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. hypot-define 100.0%

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

      8. associate-*l* 100.0%

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

      9. sqrt-prod 100.0%

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

      10. metadata-eval 100.0%

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

      11. sqrt-unprod 53.0%

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

      12. add-sqr-sqrt 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-neg-out 100.0%

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

      4. unsub-neg 100.0%

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

      5. distribute-frac-neg2 100.0%

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

      6. distribute-neg-frac 100.0%

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

      7. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around -inf 4.5%

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

      1. *-commutative 4.5%

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

      2. associate-*l/ 4.5%

         \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

      3. associate-*r/ 4.5%

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

   9. Simplified 4.5%

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

   10. Taylor expanded in p around 0 3.5%

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

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 6.4% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. +-commutative 80.0%

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

   2. frac-2neg 80.0%

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

   3. div-inv 79.3%

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

   4. fma-define 76.6%

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

   5. +-commutative 76.6%

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

   6. add-sqr-sqrt 76.6%

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

   7. hypot-define 76.6%

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

   8. associate-*l* 76.6%

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

   9. sqrt-prod 76.6%

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

   10. metadata-eval 76.6%

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

   11. sqrt-unprod 43.6%

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

   12. add-sqr-sqrt 76.6%

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

4. Applied egg-rr 76.6%

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

   1. fma-undefine 79.3%

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

   2. +-commutative 79.3%

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

   3. distribute-lft-neg-out 79.3%

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

   4. unsub-neg 79.3%

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

   5. distribute-frac-neg2 79.3%

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

   6. distribute-neg-frac 79.3%

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

   7. metadata-eval 79.3%

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

6. Simplified 79.3%

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

7. Taylor expanded in x around -inf 16.5%

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

   1. *-commutative 16.5%

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

   2. associate-*l/ 16.5%

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\frac{{p}^{2} \cdot 2}{{x}^{2}}}} \]

   3. associate-*r/ 16.5%

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

9. Simplified 16.5%

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

10. Taylor expanded in p around 0 15.2%

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

11. Final simplification 15.2%

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

Developer target: 79.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (fabs x) 1e+150))

  :alt
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

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