Given's Rotation SVD example

Percentage Accurate: 79.0% → 99.7%

Time: 13.5s

Alternatives: 7

Speedup: 0.7×

Specification

\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.0% 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, 0.3× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, p\_m \cdot 2\right)\\ \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.999996:\\ \;\;\;\;\frac{p\_m + \frac{\frac{{p\_m}^{3} \cdot -1.5}{x}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{t\_0}{x}\right)}^{-2}}, \sqrt[3]{\frac{x}{t\_0}}, 1\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (hypot x (* p_m 2.0))))
   (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999996)
     (/ (+ p_m (/ (/ (* (pow p_m 3.0) -1.5) x) x)) (- x))
     (sqrt (* (fma (cbrt (pow (/ t_0 x) -2.0)) (cbrt (/ x t_0)) 1.0) 0.5)))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = hypot(x, (p_m * 2.0));
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996) {
		tmp = (p_m + (((pow(p_m, 3.0) * -1.5) / x) / x)) / -x;
	} else {
		tmp = sqrt((fma(cbrt(pow((t_0 / x), -2.0)), cbrt((x / t_0)), 1.0) * 0.5));
	}
	return tmp;
}

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = hypot(x, Float64(p_m * 2.0))
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.999996)
		tmp = Float64(Float64(p_m + Float64(Float64(Float64((p_m ^ 3.0) * -1.5) / x) / x)) / Float64(-x));
	else
		tmp = sqrt(Float64(fma(cbrt((Float64(t_0 / x) ^ -2.0)), cbrt(Float64(x / t_0)), 1.0) * 0.5));
	end
	return tmp
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, 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], -0.999996], N[(N[(p$95$m + N[(N[(N[(N[Power[p$95$m, 3.0], $MachinePrecision] * -1.5), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(N[(N[Power[N[Power[N[(t$95$0 / x), $MachinePrecision], -2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(x / t$95$0), $MachinePrecision], 1/3], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. sqr-neg 19.6%

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

      3. associate-*l* 19.6%

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

      4. sqr-neg 19.6%

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

      5. fma-define 19.6%

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

      6. sqr-neg 19.6%

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

      7. fma-define 19.6%

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

      8. associate-*l* 19.6%

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

      9. +-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in x around -inf 47.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 47.5%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-rgt-out 47.5%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}}{x} \]

      3. metadata-eval 47.5%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}}{x} \]

   7. Simplified 47.5%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]

   8. Taylor expanded in x around inf 55.5%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 55.5%

         \[\leadsto -\frac{p + \color{blue}{\frac{{p}^{3}}{{x}^{2}} \cdot -1.5}}{x} \]

   10. Simplified 55.5%

       \[\leadsto -\color{blue}{\frac{p + \frac{{p}^{3}}{{x}^{2}} \cdot -1.5}{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 55.5%

          \[\leadsto -\frac{p + \color{blue}{\frac{{p}^{3} \cdot -1.5}{{x}^{2}}}}{x} \]

       2. unpow2 55.5%

          \[\leadsto -\frac{p + \frac{{p}^{3} \cdot -1.5}{\color{blue}{x \cdot x}}}{x} \]

       3. associate-/r* 55.5%

          \[\leadsto -\frac{p + \color{blue}{\frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}}{x} \]

   12. Applied egg-rr 55.5%

       \[\leadsto -\frac{p + \color{blue}{\frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}}{x} \]

   if -0.999995999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 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. 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. sqr-neg 99.9%

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

      3. associate-*l* 99.9%

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

      4. sqr-neg 99.9%

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

      5. fma-define 99.9%

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

      6. sqr-neg 99.9%

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

      7. fma-define 99.9%

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

      8. associate-*l* 99.9%

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

      9. +-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. fma-undefine 99.9%

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

      3. associate-*r* 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt1-in 99.9%

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

      6. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. clear-num 99.9%

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

      3. *-commutative 99.9%

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

      4. add-cube-cbrt 99.9%

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

      5. fma-define 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999996:\\ \;\;\;\;\frac{p + \frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt[3]{{\left(\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}\right)}^{-2}}, \sqrt[3]{\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)}}, 1\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.999996:\\ \;\;\;\;\frac{p\_m + \frac{\frac{{p\_m}^{3} \cdot -1.5}{x}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{\frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)} + 1}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999996)
   (/ (+ p_m (/ (/ (* (pow p_m 3.0) -1.5) x) x)) (- x))
   (sqrt (* 0.5 (log (exp (+ (/ x (hypot x (* p_m 2.0))) 1.0)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996) {
		tmp = (p_m + (((pow(p_m, 3.0) * -1.5) / x) / x)) / -x;
	} else {
		tmp = sqrt((0.5 * log(exp(((x / hypot(x, (p_m * 2.0))) + 1.0)))));
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((x / Math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996) {
		tmp = (p_m + (((Math.pow(p_m, 3.0) * -1.5) / x) / x)) / -x;
	} else {
		tmp = Math.sqrt((0.5 * Math.log(Math.exp(((x / Math.hypot(x, (p_m * 2.0))) + 1.0)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996:
		tmp = (p_m + (((math.pow(p_m, 3.0) * -1.5) / x) / x)) / -x
	else:
		tmp = math.sqrt((0.5 * math.log(math.exp(((x / math.hypot(x, (p_m * 2.0))) + 1.0)))))
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if (Float64(x / sqrt(Float64(Float64(p_m * Float64(4.0 * p_m)) + Float64(x * x)))) <= -0.999996)
		tmp = Float64(Float64(p_m + Float64(Float64(Float64((p_m ^ 3.0) * -1.5) / x) / x)) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * log(exp(Float64(Float64(x / hypot(x, Float64(p_m * 2.0))) + 1.0)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996)
		tmp = (p_m + ((((p_m ^ 3.0) * -1.5) / x) / x)) / -x;
	else
		tmp = sqrt((0.5 * log(exp(((x / hypot(x, (p_m * 2.0))) + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(p$95$m + N[(N[(N[(N[Power[p$95$m, 3.0], $MachinePrecision] * -1.5), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[Log[N[Exp[N[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. sqr-neg 19.6%

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

      3. associate-*l* 19.6%

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

      4. sqr-neg 19.6%

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

      5. fma-define 19.6%

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

      6. sqr-neg 19.6%

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

      7. fma-define 19.6%

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

      8. associate-*l* 19.6%

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

      9. +-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in x around -inf 47.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 47.5%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-rgt-out 47.5%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}}{x} \]

      3. metadata-eval 47.5%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}}{x} \]

   7. Simplified 47.5%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]

   8. Taylor expanded in x around inf 55.5%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 55.5%

         \[\leadsto -\frac{p + \color{blue}{\frac{{p}^{3}}{{x}^{2}} \cdot -1.5}}{x} \]

   10. Simplified 55.5%

       \[\leadsto -\color{blue}{\frac{p + \frac{{p}^{3}}{{x}^{2}} \cdot -1.5}{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 55.5%

          \[\leadsto -\frac{p + \color{blue}{\frac{{p}^{3} \cdot -1.5}{{x}^{2}}}}{x} \]

       2. unpow2 55.5%

          \[\leadsto -\frac{p + \frac{{p}^{3} \cdot -1.5}{\color{blue}{x \cdot x}}}{x} \]

       3. associate-/r* 55.5%

          \[\leadsto -\frac{p + \color{blue}{\frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}}{x} \]

   12. Applied egg-rr 55.5%

       \[\leadsto -\frac{p + \color{blue}{\frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}}{x} \]

   if -0.999995999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 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. add-log-exp 99.8%

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

      2. +-commutative 99.8%

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

      3. add-sqr-sqrt 99.8%

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

      4. hypot-define 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. sqrt-prod 99.9%

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

      8. sqrt-prod 50.1%

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

      9. add-sqr-sqrt 99.9%

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

      10. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999996:\\ \;\;\;\;\frac{p + \frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p\_m \cdot \left(4 \cdot p\_m\right) + x \cdot x}} \leq -0.999996:\\ \;\;\;\;\frac{p\_m + \frac{\frac{{p\_m}^{3} \cdot -1.5}{x}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\frac{\mathsf{hypot}\left(x, p\_m \cdot 2\right)}{x}}\right)}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= (/ x (sqrt (+ (* p_m (* 4.0 p_m)) (* x x)))) -0.999996)
   (/ (+ p_m (/ (/ (* (pow p_m 3.0) -1.5) x) x)) (- x))
   (sqrt (* 0.5 (+ 1.0 (/ 1.0 (/ (hypot x (* p_m 2.0)) x)))))))

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)))) <= -0.999996) {
		tmp = (p_m + (((pow(p_m, 3.0) * -1.5) / x) / x)) / -x;
	} else {
		tmp = sqrt((0.5 * (1.0 + (1.0 / (hypot(x, (p_m * 2.0)) / x)))));
	}
	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)))) <= -0.999996) {
		tmp = (p_m + (((Math.pow(p_m, 3.0) * -1.5) / x) / x)) / -x;
	} else {
		tmp = Math.sqrt((0.5 * (1.0 + (1.0 / (Math.hypot(x, (p_m * 2.0)) / x)))));
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (x / math.sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996:
		tmp = (p_m + (((math.pow(p_m, 3.0) * -1.5) / x) / x)) / -x
	else:
		tmp = math.sqrt((0.5 * (1.0 + (1.0 / (math.hypot(x, (p_m * 2.0)) / x)))))
	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)))) <= -0.999996)
		tmp = Float64(Float64(p_m + Float64(Float64(Float64((p_m ^ 3.0) * -1.5) / x) / x)) / Float64(-x));
	else
		tmp = sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / Float64(hypot(x, Float64(p_m * 2.0)) / x)))));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((x / sqrt(((p_m * (4.0 * p_m)) + (x * x)))) <= -0.999996)
		tmp = (p_m + ((((p_m ^ 3.0) * -1.5) / x) / x)) / -x;
	else
		tmp = sqrt((0.5 * (1.0 + (1.0 / (hypot(x, (p_m * 2.0)) / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[LessEqual[N[(x / N[Sqrt[N[(N[(p$95$m * N[(4.0 * p$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.999996], N[(N[(p$95$m + N[(N[(N[(N[Power[p$95$m, 3.0], $MachinePrecision] * -1.5), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[(N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. sqr-neg 19.6%

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

      3. associate-*l* 19.6%

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

      4. sqr-neg 19.6%

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

      5. fma-define 19.6%

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

      6. sqr-neg 19.6%

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

      7. fma-define 19.6%

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

      8. associate-*l* 19.6%

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

      9. +-commutative 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in x around -inf 47.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]
   6. Step-by-step derivation

      1. mul-1-neg 47.5%

         \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{-16 \cdot {p}^{4} + 4 \cdot {p}^{4}}{p \cdot {x}^{2}}}{x}} \]

      2. distribute-rgt-out 47.5%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{\color{blue}{{p}^{4} \cdot \left(-16 + 4\right)}}{p \cdot {x}^{2}}}{x} \]

      3. metadata-eval 47.5%

         \[\leadsto -\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot \color{blue}{-12}}{p \cdot {x}^{2}}}{x} \]

   7. Simplified 47.5%

      \[\leadsto \color{blue}{-\frac{p + 0.125 \cdot \frac{{p}^{4} \cdot -12}{p \cdot {x}^{2}}}{x}} \]

   8. Taylor expanded in x around inf 55.5%

      \[\leadsto -\color{blue}{\frac{p + -1.5 \cdot \frac{{p}^{3}}{{x}^{2}}}{x}} \]
   9. Step-by-step derivation

      1. *-commutative 55.5%

         \[\leadsto -\frac{p + \color{blue}{\frac{{p}^{3}}{{x}^{2}} \cdot -1.5}}{x} \]

   10. Simplified 55.5%

       \[\leadsto -\color{blue}{\frac{p + \frac{{p}^{3}}{{x}^{2}} \cdot -1.5}{x}} \]
   11. Step-by-step derivation

       1. associate-*l/ 55.5%

          \[\leadsto -\frac{p + \color{blue}{\frac{{p}^{3} \cdot -1.5}{{x}^{2}}}}{x} \]

       2. unpow2 55.5%

          \[\leadsto -\frac{p + \frac{{p}^{3} \cdot -1.5}{\color{blue}{x \cdot x}}}{x} \]

       3. associate-/r* 55.5%

          \[\leadsto -\frac{p + \color{blue}{\frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}}{x} \]

   12. Applied egg-rr 55.5%

       \[\leadsto -\frac{p + \color{blue}{\frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}}{x} \]

   if -0.999995999999999996 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 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. clear-num 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. fma-undefine 99.9%

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

      5. inv-pow 99.9%

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

      6. fma-undefine 99.9%

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

      7. associate-*r* 99.9%

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

      8. add-sqr-sqrt 99.9%

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

      9. hypot-define 99.9%

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

      10. associate-*r* 99.9%

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

      11. *-commutative 99.9%

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

      12. sqrt-prod 99.9%

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

      13. sqrt-prod 50.1%

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

      14. add-sqr-sqrt 99.9%

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

      15. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.999996:\\ \;\;\;\;\frac{p + \frac{\frac{{p}^{3} \cdot -1.5}{x}}{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \left(1 + \frac{1}{\frac{\mathsf{hypot}\left(x, p \cdot 2\right)}{x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 4 \cdot 10^{-171} \lor \neg \left(p\_m \leq 8 \cdot 10^{-108}\right):\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p\_m \cdot 2\right)} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (or (<= p_m 4e-171) (not (<= p_m 8e-108)))
   (sqrt (* 0.5 (+ (/ x (hypot x (* p_m 2.0))) 1.0)))
   (/ p_m (- x))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if ((p_m <= 4e-171) || !(p_m <= 8e-108)) {
		tmp = sqrt((0.5 * ((x / hypot(x, (p_m * 2.0))) + 1.0)));
	} else {
		tmp = p_m / -x;
	}
	return tmp;
}

Java

p_m = Math.abs(p);
public static double code(double p_m, double x) {
	double tmp;
	if ((p_m <= 4e-171) || !(p_m <= 8e-108)) {
		tmp = Math.sqrt((0.5 * ((x / Math.hypot(x, (p_m * 2.0))) + 1.0)));
	} else {
		tmp = p_m / -x;
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	tmp = 0
	if (p_m <= 4e-171) or not (p_m <= 8e-108):
		tmp = math.sqrt((0.5 * ((x / math.hypot(x, (p_m * 2.0))) + 1.0)))
	else:
		tmp = p_m / -x
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	tmp = 0.0
	if ((p_m <= 4e-171) || !(p_m <= 8e-108))
		tmp = sqrt(Float64(0.5 * Float64(Float64(x / hypot(x, Float64(p_m * 2.0))) + 1.0)));
	else
		tmp = Float64(p_m / Float64(-x));
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	tmp = 0.0;
	if ((p_m <= 4e-171) || ~((p_m <= 8e-108)))
		tmp = sqrt((0.5 * ((x / hypot(x, (p_m * 2.0))) + 1.0)));
	else
		tmp = p_m / -x;
	end
	tmp_2 = tmp;
end

Wolfram

p_m = N[Abs[p], $MachinePrecision]
code[p$95$m_, x_] := If[Or[LessEqual[p$95$m, 4e-171], N[Not[LessEqual[p$95$m, 8e-108]], $MachinePrecision]], N[Sqrt[N[(0.5 * N[(N[(x / N[Sqrt[x ^ 2 + N[(p$95$m * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(p$95$m / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 3.9999999999999999e-171 or 8.00000000000000032e-108 < p

   1. Initial program 81.9%

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

      1. +-commutative 81.9%

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

      2. sqr-neg 81.9%

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

      3. associate-*l* 81.9%

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

      4. sqr-neg 81.9%

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

      5. fma-define 81.9%

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

      6. sqr-neg 81.9%

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

      7. fma-define 81.9%

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

      8. associate-*l* 81.9%

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

      9. +-commutative 81.9%

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

   3. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. fma-undefine 81.9%

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

      3. associate-*r* 81.9%

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

      4. +-commutative 81.9%

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

      5. distribute-rgt1-in 81.9%

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

      6. +-commutative 81.9%

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

   6. Applied egg-rr 81.9%

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

   if 3.9999999999999999e-171 < p < 8.00000000000000032e-108

   1. Initial program 25.4%

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

      1. +-commutative 25.4%

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

      2. sqr-neg 25.4%

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

      3. associate-*l* 25.4%

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

      4. sqr-neg 25.4%

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

      5. fma-define 25.4%

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

      6. sqr-neg 25.4%

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

      7. fma-define 25.4%

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

      8. associate-*l* 25.4%

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

      9. +-commutative 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in x around -inf 79.1%

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

      1. mul-1-neg 79.1%

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

      2. distribute-neg-frac2 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4 \cdot 10^{-171} \lor \neg \left(p \leq 8 \cdot 10^{-108}\right):\\ \;\;\;\;\sqrt{0.5 \cdot \left(\frac{x}{\mathsf{hypot}\left(x, p \cdot 2\right)} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{p}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} t_0 := \frac{p\_m}{-x}\\ \mathbf{if}\;p\_m \leq 1.05 \cdot 10^{-255}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 7.6 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;p\_m \leq 5.5 \cdot 10^{-110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;p\_m \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (let* ((t_0 (/ p_m (- x))))
   (if (<= p_m 1.05e-255)
     t_0
     (if (<= p_m 7.6e-171)
       1.0
       (if (<= p_m 5.5e-110) t_0 (if (<= p_m 1.6e-42) 1.0 (sqrt 0.5)))))))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double t_0 = p_m / -x;
	double tmp;
	if (p_m <= 1.05e-255) {
		tmp = t_0;
	} else if (p_m <= 7.6e-171) {
		tmp = 1.0;
	} else if (p_m <= 5.5e-110) {
		tmp = t_0;
	} else if (p_m <= 1.6e-42) {
		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) :: t_0
    real(8) :: tmp
    t_0 = p_m / -x
    if (p_m <= 1.05d-255) then
        tmp = t_0
    else if (p_m <= 7.6d-171) then
        tmp = 1.0d0
    else if (p_m <= 5.5d-110) then
        tmp = t_0
    else if (p_m <= 1.6d-42) 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 t_0 = p_m / -x;
	double tmp;
	if (p_m <= 1.05e-255) {
		tmp = t_0;
	} else if (p_m <= 7.6e-171) {
		tmp = 1.0;
	} else if (p_m <= 5.5e-110) {
		tmp = t_0;
	} else if (p_m <= 1.6e-42) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

p_m = math.fabs(p)
def code(p_m, x):
	t_0 = p_m / -x
	tmp = 0
	if p_m <= 1.05e-255:
		tmp = t_0
	elif p_m <= 7.6e-171:
		tmp = 1.0
	elif p_m <= 5.5e-110:
		tmp = t_0
	elif p_m <= 1.6e-42:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

p_m = abs(p)
function code(p_m, x)
	t_0 = Float64(p_m / Float64(-x))
	tmp = 0.0
	if (p_m <= 1.05e-255)
		tmp = t_0;
	elseif (p_m <= 7.6e-171)
		tmp = 1.0;
	elseif (p_m <= 5.5e-110)
		tmp = t_0;
	elseif (p_m <= 1.6e-42)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

p_m = abs(p);
function tmp_2 = code(p_m, x)
	t_0 = p_m / -x;
	tmp = 0.0;
	if (p_m <= 1.05e-255)
		tmp = t_0;
	elseif (p_m <= 7.6e-171)
		tmp = 1.0;
	elseif (p_m <= 5.5e-110)
		tmp = t_0;
	elseif (p_m <= 1.6e-42)
		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_] := Block[{t$95$0 = N[(p$95$m / (-x)), $MachinePrecision]}, If[LessEqual[p$95$m, 1.05e-255], t$95$0, If[LessEqual[p$95$m, 7.6e-171], 1.0, If[LessEqual[p$95$m, 5.5e-110], t$95$0, If[LessEqual[p$95$m, 1.6e-42], 1.0, N[Sqrt[0.5], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if p < 1.05e-255 or 7.60000000000000043e-171 < p < 5.4999999999999998e-110

   1. Initial program 72.0%

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

      1. +-commutative 72.0%

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

      2. sqr-neg 72.0%

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

      3. associate-*l* 72.0%

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

      4. sqr-neg 72.0%

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

      5. fma-define 72.0%

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

      6. sqr-neg 72.0%

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

      7. fma-define 72.0%

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

      8. associate-*l* 72.0%

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

      9. +-commutative 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in x around -inf 17.7%

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

      1. mul-1-neg 17.7%

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

      2. distribute-neg-frac2 17.7%

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

   7. Simplified 17.7%

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

   if 1.05e-255 < p < 7.60000000000000043e-171 or 5.4999999999999998e-110 < p < 1.60000000000000012e-42

   1. Initial program 71.2%

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

   3. Taylor expanded in x around inf 59.6%

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

   if 1.60000000000000012e-42 < p

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0 84.2%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 1.05 \cdot 10^{-255}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 7.6 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{elif}\;p \leq 5.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{elif}\;p \leq 1.6 \cdot 10^{-42}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} p_m = \left|p\right| \\ \begin{array}{l} \mathbf{if}\;p\_m \leq 4.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{p\_m}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

p_m = (fabs.f64 p)
(FPCore (p_m x)
 :precision binary64
 (if (<= p_m 4.2e-69) (/ p_m (- x)) (sqrt 0.5)))

C

p_m = fabs(p);
double code(double p_m, double x) {
	double tmp;
	if (p_m <= 4.2e-69) {
		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 <= 4.2d-69) 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 <= 4.2e-69) {
		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 <= 4.2e-69:
		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 <= 4.2e-69)
		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 <= 4.2e-69)
		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, 4.2e-69], N[(p$95$m / (-x)), $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if p < 4.1999999999999999e-69

   1. Initial program 72.2%

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

      1. +-commutative 72.2%

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

      2. sqr-neg 72.2%

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

      3. associate-*l* 72.2%

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

      4. sqr-neg 72.2%

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

      5. fma-define 72.2%

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

      6. sqr-neg 72.2%

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

      7. fma-define 72.2%

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

      8. associate-*l* 72.2%

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

      9. +-commutative 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in x around -inf 20.3%

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

      1. mul-1-neg 20.3%

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

      2. distribute-neg-frac2 20.3%

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

   7. Simplified 20.3%

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

   if 4.1999999999999999e-69 < p

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0 79.3%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;p \leq 4.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{p}{-x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 27.1% accurate, 53.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

p_m = abs(p)
function code(p_m, x)
	return Float64(p_m / Float64(-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 78.8%

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

   1. +-commutative 78.8%

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

   2. sqr-neg 78.8%

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

   3. associate-*l* 78.8%

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

   4. sqr-neg 78.8%

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

   5. fma-define 78.8%

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

   6. sqr-neg 78.8%

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

   7. fma-define 78.8%

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

   8. associate-*l* 78.8%

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

   9. +-commutative 78.8%

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

3. Simplified 78.8%

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

5. Taylor expanded in x around -inf 17.1%

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

   1. mul-1-neg 17.1%

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

   2. distribute-neg-frac2 17.1%

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

7. Simplified 17.1%

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

8. Final simplification 17.1%

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

Developer target: 78.9% 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 2024055 
(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))))))))