Toniolo and Linder, Equation (7)

Percentage Accurate: 33.8% → 85.0%

Time: 21.9s

Alternatives: 16

Speedup: 225.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 33.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \left(t + \frac{t}{x}\right)\\ t_2 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_3 := \frac{t_2}{x}\\ t_4 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}{t}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(t_4 + 2 \cdot t_1\right) + t_3}{2}}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{t_4 + \left(\mathsf{fma}\left(2, t_1, \frac{2 \cdot t_2}{x \cdot x}\right) + t_3\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (+ t (/ t x))))
        (t_2 (fma 2.0 (* t t) (* l l)))
        (t_3 (/ t_2 x))
        (t_4 (/ (* l l) x)))
   (if (<= t -9.2e+49)
     (/
      (sqrt 2.0)
      (/ (* (* t (sqrt 2.0)) (- (sqrt (/ (+ x 1.0) (+ x -1.0))))) t))
     (if (<= t -5e-158)
       (/ t (sqrt (/ (+ (+ t_4 (* 2.0 t_1)) t_3) 2.0)))
       (if (<= t 1.15e-161)
         (*
          t
          (/
           (sqrt 2.0)
           (* l (sqrt (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) (/ 2.0 (pow x 3.0))))))))
         (if (<= t 4.8e+44)
           (/
            t
            (sqrt
             (/ (+ t_4 (+ (fma 2.0 t_1 (/ (* 2.0 t_2) (* x x))) t_3)) 2.0)))
           (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (t + (t / x));
	double t_2 = fma(2.0, (t * t), (l * l));
	double t_3 = t_2 / x;
	double t_4 = (l * l) / x;
	double tmp;
	if (t <= -9.2e+49) {
		tmp = sqrt(2.0) / (((t * sqrt(2.0)) * -sqrt(((x + 1.0) / (x + -1.0)))) / t);
	} else if (t <= -5e-158) {
		tmp = t / sqrt((((t_4 + (2.0 * t_1)) + t_3) / 2.0));
	} else if (t <= 1.15e-161) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + ((2.0 / (x * x)) + (2.0 / pow(x, 3.0)))))));
	} else if (t <= 4.8e+44) {
		tmp = t / sqrt(((t_4 + (fma(2.0, t_1, ((2.0 * t_2) / (x * x))) + t_3)) / 2.0));
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(t + Float64(t / x)))
	t_2 = fma(2.0, Float64(t * t), Float64(l * l))
	t_3 = Float64(t_2 / x)
	t_4 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -9.2e+49)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(t * sqrt(2.0)) * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))) / t));
	elseif (t <= -5e-158)
		tmp = Float64(t / sqrt(Float64(Float64(Float64(t_4 + Float64(2.0 * t_1)) + t_3) / 2.0)));
	elseif (t <= 1.15e-161)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / (x ^ 3.0))))))));
	elseif (t <= 4.8e+44)
		tmp = Float64(t / sqrt(Float64(Float64(t_4 + Float64(fma(2.0, t_1, Float64(Float64(2.0 * t_2) / Float64(x * x))) + t_3)) / 2.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / x), $MachinePrecision]}, Block[{t$95$4 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -9.2e+49], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-158], N[(t / N[Sqrt[N[(N[(N[(t$95$4 + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-161], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e+44], N[(t / N[Sqrt[N[(N[(t$95$4 + N[(N[(2.0 * t$95$1 + N[(N[(2.0 * t$95$2), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -9.20000000000000008e49

   1. Initial program 27.0%

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

      1. associate-*l/ 27.1%

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

   3. Simplified 27.1%

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

      1. associate-/r/ 27.1%

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

      2. fma-neg 27.1%

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

      3. metadata-eval 27.1%

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

      4. sub-neg 27.1%

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

      5. fma-udef 27.1%

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

      6. +-commutative 27.1%

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

      7. fma-neg 27.1%

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

   5. Applied egg-rr 12.2%

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

   6. Taylor expanded in t around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. *-commutative 92.9%

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

      4. +-commutative 92.9%

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

   8. Simplified 92.9%

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

   if -9.20000000000000008e49 < t < -4.99999999999999972e-158

   1. Initial program 57.3%

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

      1. associate-*l/ 57.5%

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

   3. Simplified 57.5%

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

      1. *-commutative 57.5%

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

      2. clear-num 57.4%

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

      3. un-div-inv 57.3%

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

      4. sqrt-undiv 57.7%

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

      5. metadata-eval 57.7%

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

      6. sub-neg 57.7%

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

      7. associate-*l/ 55.6%

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

      8. sub-neg 55.6%

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

      9. metadata-eval 55.6%

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

   5. Applied egg-rr 55.6%

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

      1. +-commutative 55.6%

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

      2. *-commutative 55.6%

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

      3. +-commutative 55.6%

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

      4. +-commutative 55.6%

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

   7. Simplified 55.6%

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

   8. Taylor expanded in x around inf 92.2%

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

      1. unpow2 92.2%

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

      2. +-commutative 92.2%

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

      3. distribute-lft-out 92.2%

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

      4. unpow2 92.2%

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

      5. unpow2 92.2%

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

      6. associate-*l/ 92.2%

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

      7. distribute-rgt-out 92.2%

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

      8. associate-*r/ 92.2%

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

      9. mul-1-neg 92.2%

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

      10. +-commutative 92.2%

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

      11. unpow2 92.2%

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

      12. unpow2 92.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 92.2%

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

   10. Simplified 92.2%

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

   if -4.99999999999999972e-158 < t < 1.15e-161

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 7.2%

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

      1. *-commutative 7.2%

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

      2. associate--l+ 7.3%

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

      3. sub-neg 7.3%

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

      4. metadata-eval 7.3%

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

      5. +-commutative 7.3%

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

      6. sub-neg 7.3%

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

      7. metadata-eval 7.3%

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

      8. +-commutative 7.3%

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

   6. Simplified 7.3%

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

   7. Taylor expanded in x around inf 57.3%

      \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{x} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}}} \cdot t \]
   8. Step-by-step derivation

      1. associate-*r/ 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{\frac{2 \cdot 1}{x}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}} \cdot t \]

      2. metadata-eval 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{\color{blue}{2}}{x} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}} \cdot t \]

      3. +-commutative 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{3}}\right)}}} \cdot t \]

      4. associate-*r/ 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 2 \cdot \frac{1}{{x}^{3}}\right)}} \cdot t \]

      5. metadata-eval 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{\color{blue}{2}}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{3}}\right)}} \cdot t \]

      6. unpow2 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{{x}^{3}}\right)}} \cdot t \]

      7. associate-*r/ 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}}\right)}} \cdot t \]

      8. metadata-eval 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{{x}^{3}}\right)}} \cdot t \]

   9. Simplified 57.3%

      \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}} \cdot t \]

   if 1.15e-161 < t < 4.80000000000000026e44

   1. Initial program 54.8%

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

      1. associate-*l/ 55.0%

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

   3. Simplified 55.0%

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

      1. *-commutative 55.0%

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

      2. clear-num 54.9%

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

      3. un-div-inv 54.7%

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

      4. sqrt-undiv 55.1%

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

      5. metadata-eval 55.1%

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

      6. sub-neg 55.1%

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

      7. associate-*l/ 52.7%

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

      8. sub-neg 52.7%

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

      9. metadata-eval 52.7%

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

   5. Applied egg-rr 52.7%

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

      1. +-commutative 52.7%

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

      2. *-commutative 52.7%

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

      3. +-commutative 52.7%

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

      4. +-commutative 52.7%

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

   7. Simplified 52.7%

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

   8. Taylor expanded in x around -inf 88.2%

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

   10. Simplified 88.2%

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

   if 4.80000000000000026e44 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

       \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}{t}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, t \cdot \left(t + \frac{t}{x}\right), \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 2: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{if}\;t \leq -3.25 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}{t}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (+
             (+ (/ (* l l) x) (* 2.0 (* t (+ t (/ t x)))))
             (/ (fma 2.0 (* t t) (* l l)) x))
            2.0)))))
   (if (<= t -3.25e+50)
     (/
      (sqrt 2.0)
      (/ (* (* t (sqrt 2.0)) (- (sqrt (/ (+ x 1.0) (+ x -1.0))))) t))
     (if (<= t -4.5e-158)
       t_1
       (if (<= t 1.6e-162)
         (*
          t
          (/
           (sqrt 2.0)
           (* l (sqrt (+ (/ 2.0 x) (+ (/ 2.0 (* x x)) (/ 2.0 (pow x 3.0))))))))
         (if (<= t 1.35e+45)
           t_1
           (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / sqrt((((((l * l) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, (t * t), (l * l)) / x)) / 2.0));
	double tmp;
	if (t <= -3.25e+50) {
		tmp = sqrt(2.0) / (((t * sqrt(2.0)) * -sqrt(((x + 1.0) / (x + -1.0)))) / t);
	} else if (t <= -4.5e-158) {
		tmp = t_1;
	} else if (t <= 1.6e-162) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + ((2.0 / (x * x)) + (2.0 / pow(x, 3.0)))))));
	} else if (t <= 1.35e+45) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)) / 2.0)))
	tmp = 0.0
	if (t <= -3.25e+50)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(t * sqrt(2.0)) * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))) / t));
	elseif (t <= -4.5e-158)
		tmp = t_1;
	elseif (t <= 1.6e-162)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / (x ^ 3.0))))))));
	elseif (t <= 1.35e+45)
		tmp = t_1;
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.25e+50], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-158], t$95$1, If[LessEqual[t, 1.6e-162], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e+45], t$95$1, N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.2500000000000001e50

   1. Initial program 27.0%

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

      1. associate-*l/ 27.1%

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

   3. Simplified 27.1%

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

      1. associate-/r/ 27.1%

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

      2. fma-neg 27.1%

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

      3. metadata-eval 27.1%

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

      4. sub-neg 27.1%

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

      5. fma-udef 27.1%

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

      6. +-commutative 27.1%

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

      7. fma-neg 27.1%

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

   5. Applied egg-rr 12.2%

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

   6. Taylor expanded in t around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. *-commutative 92.9%

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

      4. +-commutative 92.9%

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

   8. Simplified 92.9%

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

   if -3.2500000000000001e50 < t < -4.5e-158 or 1.59999999999999988e-162 < t < 1.34999999999999992e45

   1. Initial program 56.2%

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

      1. associate-*l/ 56.4%

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

   3. Simplified 56.4%

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

      1. *-commutative 56.4%

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

      2. clear-num 56.2%

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

      3. un-div-inv 56.1%

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

      4. sqrt-undiv 56.5%

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

      5. metadata-eval 56.5%

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

      6. sub-neg 56.5%

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

      7. associate-*l/ 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

   5. Applied egg-rr 54.3%

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

      1. +-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. +-commutative 54.3%

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

      4. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 90.2%

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

      1. unpow2 90.2%

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

      2. +-commutative 90.2%

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

      3. distribute-lft-out 90.2%

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

      4. unpow2 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 90.2%

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

      7. distribute-rgt-out 90.2%

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

      8. associate-*r/ 90.2%

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

      9. mul-1-neg 90.2%

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

      10. +-commutative 90.2%

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

      11. unpow2 90.2%

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

      12. unpow2 90.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 90.2%

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

   10. Simplified 90.2%

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

   if -4.5e-158 < t < 1.59999999999999988e-162

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in l around inf 7.2%

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

      1. *-commutative 7.2%

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

      2. associate--l+ 7.3%

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

      3. sub-neg 7.3%

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

      4. metadata-eval 7.3%

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

      5. +-commutative 7.3%

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

      6. sub-neg 7.3%

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

      7. metadata-eval 7.3%

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

      8. +-commutative 7.3%

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

   6. Simplified 7.3%

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

   7. Taylor expanded in x around inf 57.3%

      \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{2 \cdot \frac{1}{x} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}}} \cdot t \]
   8. Step-by-step derivation

      1. associate-*r/ 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{\frac{2 \cdot 1}{x}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}} \cdot t \]

      2. metadata-eval 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{\color{blue}{2}}{x} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{2}}\right)}} \cdot t \]

      3. +-commutative 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \color{blue}{\left(2 \cdot \frac{1}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{3}}\right)}}} \cdot t \]

      4. associate-*r/ 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\color{blue}{\frac{2 \cdot 1}{{x}^{2}}} + 2 \cdot \frac{1}{{x}^{3}}\right)}} \cdot t \]

      5. metadata-eval 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{\color{blue}{2}}{{x}^{2}} + 2 \cdot \frac{1}{{x}^{3}}\right)}} \cdot t \]

      6. unpow2 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{{x}^{3}}\right)}} \cdot t \]

      7. associate-*r/ 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{{x}^{3}}}\right)}} \cdot t \]

      8. metadata-eval 57.3%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{{x}^{3}}\right)}} \cdot t \]

   9. Simplified 57.3%

      \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}} \cdot t \]

   if 1.34999999999999992e45 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.25 \cdot 10^{+50}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}{t}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \left(\frac{2}{x \cdot x} + \frac{2}{{x}^{3}}\right)}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+45}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 3: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{x + -1}{x + 1}\\ t_2 := \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \left(\sqrt{t_1} \cdot \frac{-\sqrt{2}}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+44}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot t_1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (+ x -1.0) (+ x 1.0)))
        (t_2
         (/
          t
          (sqrt
           (/
            (+
             (+ (/ (* l l) x) (* 2.0 (* t (+ t (/ t x)))))
             (/ (fma 2.0 (* t t) (* l l)) x))
            2.0)))))
   (if (<= t -1.65e+50)
     (* t (* (sqrt t_1) (/ (- (sqrt 2.0)) (/ t (sqrt 0.5)))))
     (if (<= t -4.5e-158)
       t_2
       (if (<= t 1.75e-162)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 3.3e+44) t_2 (sqrt (* 2.0 (* 0.5 t_1)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (x + -1.0) / (x + 1.0);
	double t_2 = t / sqrt((((((l * l) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, (t * t), (l * l)) / x)) / 2.0));
	double tmp;
	if (t <= -1.65e+50) {
		tmp = t * (sqrt(t_1) * (-sqrt(2.0) / (t / sqrt(0.5))));
	} else if (t <= -4.5e-158) {
		tmp = t_2;
	} else if (t <= 1.75e-162) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 3.3e+44) {
		tmp = t_2;
	} else {
		tmp = sqrt((2.0 * (0.5 * t_1)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	t_2 = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)) / 2.0)))
	tmp = 0.0
	if (t <= -1.65e+50)
		tmp = Float64(t * Float64(sqrt(t_1) * Float64(Float64(-sqrt(2.0)) / Float64(t / sqrt(0.5)))));
	elseif (t <= -4.5e-158)
		tmp = t_2;
	elseif (t <= 1.75e-162)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 3.3e+44)
		tmp = t_2;
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * t_1)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.65e+50], N[(t * N[(N[Sqrt[t$95$1], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-158], t$95$2, If[LessEqual[t, 1.75e-162], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e+44], t$95$2, N[Sqrt[N[(2.0 * N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.65e50

   1. Initial program 27.0%

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

      1. associate-*l/ 27.1%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in t around -inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. *-commutative 91.5%

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

      3. distribute-rgt-neg-in 91.5%

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

      4. sub-neg 91.5%

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

      5. metadata-eval 91.5%

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

      6. +-commutative 91.5%

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

      7. +-commutative 91.5%

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

      8. associate-/l* 91.9%

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

   6. Simplified 91.9%

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

   if -1.65e50 < t < -4.5e-158 or 1.74999999999999995e-162 < t < 3.30000000000000013e44

   1. Initial program 56.2%

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

      1. associate-*l/ 56.4%

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

   3. Simplified 56.4%

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

      1. *-commutative 56.4%

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

      2. clear-num 56.2%

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

      3. un-div-inv 56.1%

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

      4. sqrt-undiv 56.5%

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

      5. metadata-eval 56.5%

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

      6. sub-neg 56.5%

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

      7. associate-*l/ 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

   5. Applied egg-rr 54.3%

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

      1. +-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. +-commutative 54.3%

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

      4. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 90.2%

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

      1. unpow2 90.2%

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

      2. +-commutative 90.2%

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

      3. distribute-lft-out 90.2%

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

      4. unpow2 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 90.2%

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

      7. distribute-rgt-out 90.2%

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

      8. associate-*r/ 90.2%

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

      9. mul-1-neg 90.2%

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

      10. +-commutative 90.2%

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

      11. unpow2 90.2%

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

      12. unpow2 90.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 90.2%

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

   10. Simplified 90.2%

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

   if -4.5e-158 < t < 1.74999999999999995e-162

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around -inf 48.4%

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

      1. associate--l+ 48.4%

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

      2. unpow2 48.4%

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

      3. sub-neg 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in l around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. unpow2 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 3.30000000000000013e44 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \frac{-\sqrt{2}}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 4: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (+
             (+ (/ (* l l) x) (* 2.0 (* t (+ t (/ t x)))))
             (/ (fma 2.0 (* t t) (* l l)) x))
            2.0)))))
   (if (<= t -1.65e+50)
     (*
      t
      (/ (sqrt 2.0) (* (* t (sqrt 2.0)) (- (sqrt (/ (+ x 1.0) (+ x -1.0)))))))
     (if (<= t -4.5e-158)
       t_1
       (if (<= t 1.28e-163)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 1e+45)
           t_1
           (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / sqrt((((((l * l) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, (t * t), (l * l)) / x)) / 2.0));
	double tmp;
	if (t <= -1.65e+50) {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * -sqrt(((x + 1.0) / (x + -1.0)))));
	} else if (t <= -4.5e-158) {
		tmp = t_1;
	} else if (t <= 1.28e-163) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 1e+45) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)) / 2.0)))
	tmp = 0.0
	if (t <= -1.65e+50)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0)))))));
	elseif (t <= -4.5e-158)
		tmp = t_1;
	elseif (t <= 1.28e-163)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 1e+45)
		tmp = t_1;
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.65e+50], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-158], t$95$1, If[LessEqual[t, 1.28e-163], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+45], t$95$1, N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.65e50

   1. Initial program 27.0%

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

      1. associate-*l/ 27.1%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

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

      2. *-commutative 92.5%

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

      3. distribute-rgt-neg-in 92.5%

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

      4. +-commutative 92.5%

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

      5. sub-neg 92.5%

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

      6. metadata-eval 92.5%

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

      7. +-commutative 92.5%

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

      8. distribute-rgt-neg-out 92.5%

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

   6. Simplified 92.5%

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

   if -1.65e50 < t < -4.5e-158 or 1.28e-163 < t < 9.9999999999999993e44

   1. Initial program 56.2%

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

      1. associate-*l/ 56.4%

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

   3. Simplified 56.4%

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

      1. *-commutative 56.4%

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

      2. clear-num 56.2%

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

      3. un-div-inv 56.1%

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

      4. sqrt-undiv 56.5%

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

      5. metadata-eval 56.5%

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

      6. sub-neg 56.5%

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

      7. associate-*l/ 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

   5. Applied egg-rr 54.3%

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

      1. +-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. +-commutative 54.3%

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

      4. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 90.2%

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

      1. unpow2 90.2%

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

      2. +-commutative 90.2%

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

      3. distribute-lft-out 90.2%

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

      4. unpow2 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 90.2%

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

      7. distribute-rgt-out 90.2%

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

      8. associate-*r/ 90.2%

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

      9. mul-1-neg 90.2%

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

      10. +-commutative 90.2%

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

      11. unpow2 90.2%

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

      12. unpow2 90.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 90.2%

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

   10. Simplified 90.2%

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

   if -4.5e-158 < t < 1.28e-163

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around -inf 48.4%

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

      1. associate--l+ 48.4%

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

      2. unpow2 48.4%

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

      3. sub-neg 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in l around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. unpow2 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 9.9999999999999993e44 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 10^{+45}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 5: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{if}\;t \leq -7.3 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}{t}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (+
             (+ (/ (* l l) x) (* 2.0 (* t (+ t (/ t x)))))
             (/ (fma 2.0 (* t t) (* l l)) x))
            2.0)))))
   (if (<= t -7.3e+49)
     (/
      (sqrt 2.0)
      (/ (* (* t (sqrt 2.0)) (- (sqrt (/ (+ x 1.0) (+ x -1.0))))) t))
     (if (<= t -4.5e-158)
       t_1
       (if (<= t 1.55e-162)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 2.5e+44)
           t_1
           (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / sqrt((((((l * l) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, (t * t), (l * l)) / x)) / 2.0));
	double tmp;
	if (t <= -7.3e+49) {
		tmp = sqrt(2.0) / (((t * sqrt(2.0)) * -sqrt(((x + 1.0) / (x + -1.0)))) / t);
	} else if (t <= -4.5e-158) {
		tmp = t_1;
	} else if (t <= 1.55e-162) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 2.5e+44) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)) / 2.0)))
	tmp = 0.0
	if (t <= -7.3e+49)
		tmp = Float64(sqrt(2.0) / Float64(Float64(Float64(t * sqrt(2.0)) * Float64(-sqrt(Float64(Float64(x + 1.0) / Float64(x + -1.0))))) / t));
	elseif (t <= -4.5e-158)
		tmp = t_1;
	elseif (t <= 1.55e-162)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 2.5e+44)
		tmp = t_1;
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.3e+49], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-158], t$95$1, If[LessEqual[t, 1.55e-162], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e+44], t$95$1, N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.30000000000000014e49

   1. Initial program 27.0%

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

      1. associate-*l/ 27.1%

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

   3. Simplified 27.1%

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

      1. associate-/r/ 27.1%

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

      2. fma-neg 27.1%

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

      3. metadata-eval 27.1%

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

      4. sub-neg 27.1%

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

      5. fma-udef 27.1%

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

      6. +-commutative 27.1%

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

      7. fma-neg 27.1%

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

   5. Applied egg-rr 12.2%

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

   6. Taylor expanded in t around -inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

      3. *-commutative 92.9%

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

      4. +-commutative 92.9%

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

   8. Simplified 92.9%

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

   if -7.30000000000000014e49 < t < -4.5e-158 or 1.5499999999999999e-162 < t < 2.4999999999999998e44

   1. Initial program 56.2%

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

      1. associate-*l/ 56.4%

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

   3. Simplified 56.4%

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

      1. *-commutative 56.4%

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

      2. clear-num 56.2%

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

      3. un-div-inv 56.1%

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

      4. sqrt-undiv 56.5%

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

      5. metadata-eval 56.5%

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

      6. sub-neg 56.5%

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

      7. associate-*l/ 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

   5. Applied egg-rr 54.3%

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

      1. +-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. +-commutative 54.3%

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

      4. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 90.2%

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

      1. unpow2 90.2%

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

      2. +-commutative 90.2%

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

      3. distribute-lft-out 90.2%

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

      4. unpow2 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 90.2%

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

      7. distribute-rgt-out 90.2%

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

      8. associate-*r/ 90.2%

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

      9. mul-1-neg 90.2%

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

      10. +-commutative 90.2%

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

      11. unpow2 90.2%

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

      12. unpow2 90.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 90.2%

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

   10. Simplified 90.2%

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

   if -4.5e-158 < t < 1.5499999999999999e-162

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around -inf 48.4%

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

      1. associate--l+ 48.4%

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

      2. unpow2 48.4%

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

      3. sub-neg 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in l around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. unpow2 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 2.4999999999999998e44 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.3 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{x + -1}}\right)}{t}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 6: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ t_2 := \frac{x + -1}{x + 1}\\ \mathbf{if}\;t \leq -3.25 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{t_2} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot t_2\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (+
             (+ (/ (* l l) x) (* 2.0 (* t (+ t (/ t x)))))
             (/ (fma 2.0 (* t t) (* l l)) x))
            2.0))))
        (t_2 (/ (+ x -1.0) (+ x 1.0))))
   (if (<= t -3.25e+50)
     (* (sqrt 2.0) (* (sqrt t_2) (- (sqrt 0.5))))
     (if (<= t -4.5e-158)
       t_1
       (if (<= t 1.95e-162)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 6e+44) t_1 (sqrt (* 2.0 (* 0.5 t_2)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / sqrt((((((l * l) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, (t * t), (l * l)) / x)) / 2.0));
	double t_2 = (x + -1.0) / (x + 1.0);
	double tmp;
	if (t <= -3.25e+50) {
		tmp = sqrt(2.0) * (sqrt(t_2) * -sqrt(0.5));
	} else if (t <= -4.5e-158) {
		tmp = t_1;
	} else if (t <= 1.95e-162) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 6e+44) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * (0.5 * t_2)));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)) / 2.0)))
	t_2 = Float64(Float64(x + -1.0) / Float64(x + 1.0))
	tmp = 0.0
	if (t <= -3.25e+50)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(t_2) * Float64(-sqrt(0.5))));
	elseif (t <= -4.5e-158)
		tmp = t_1;
	elseif (t <= 1.95e-162)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 6e+44)
		tmp = t_1;
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * t_2)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.25e+50], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[t$95$2], $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-158], t$95$1, If[LessEqual[t, 1.95e-162], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+44], t$95$1, N[Sqrt[N[(2.0 * N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -3.2500000000000001e50

   1. Initial program 27.0%

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

      1. associate-*l/ 27.1%

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

   3. Simplified 27.1%

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

   4. Taylor expanded in t around -inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. associate-*l* 91.5%

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

      3. sub-neg 91.5%

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

      4. metadata-eval 91.5%

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

      5. +-commutative 91.5%

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

      6. +-commutative 91.5%

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

   6. Simplified 91.5%

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

   if -3.2500000000000001e50 < t < -4.5e-158 or 1.95e-162 < t < 5.99999999999999974e44

   1. Initial program 56.2%

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

      1. associate-*l/ 56.4%

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

   3. Simplified 56.4%

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

      1. *-commutative 56.4%

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

      2. clear-num 56.2%

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

      3. un-div-inv 56.1%

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

      4. sqrt-undiv 56.5%

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

      5. metadata-eval 56.5%

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

      6. sub-neg 56.5%

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

      7. associate-*l/ 54.3%

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

      8. sub-neg 54.3%

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

      9. metadata-eval 54.3%

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

   5. Applied egg-rr 54.3%

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

      1. +-commutative 54.3%

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

      2. *-commutative 54.3%

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

      3. +-commutative 54.3%

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

      4. +-commutative 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in x around inf 90.2%

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

      1. unpow2 90.2%

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

      2. +-commutative 90.2%

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

      3. distribute-lft-out 90.2%

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

      4. unpow2 90.2%

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

      5. unpow2 90.2%

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

      6. associate-*l/ 90.2%

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

      7. distribute-rgt-out 90.2%

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

      8. associate-*r/ 90.2%

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

      9. mul-1-neg 90.2%

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

      10. +-commutative 90.2%

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

      11. unpow2 90.2%

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

      12. unpow2 90.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 90.2%

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

   10. Simplified 90.2%

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

   if -4.5e-158 < t < 1.95e-162

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around -inf 48.4%

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

      1. associate--l+ 48.4%

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

      2. unpow2 48.4%

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

      3. sub-neg 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in l around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. unpow2 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 5.99999999999999974e44 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.25 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{\frac{x + -1}{x + 1}} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 7: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (+
             (+ (/ (* l l) x) (* 2.0 (* t (+ t (/ t x)))))
             (/ (fma 2.0 (* t t) (* l l)) x))
            2.0)))))
   (if (<= t -4.6e+119)
     (/ (sqrt 2.0) (- (sqrt 2.0)))
     (if (<= t -4.5e-158)
       t_1
       (if (<= t 2.9e-162)
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
         (if (<= t 6.6e+44)
           t_1
           (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / sqrt((((((l * l) / x) + (2.0 * (t * (t + (t / x))))) + (fma(2.0, (t * t), (l * l)) / x)) / 2.0));
	double tmp;
	if (t <= -4.6e+119) {
		tmp = sqrt(2.0) / -sqrt(2.0);
	} else if (t <= -4.5e-158) {
		tmp = t_1;
	} else if (t <= 2.9e-162) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else if (t <= 6.6e+44) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(l * l) / x) + Float64(2.0 * Float64(t * Float64(t + Float64(t / x))))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)) / 2.0)))
	tmp = 0.0
	if (t <= -4.6e+119)
		tmp = Float64(sqrt(2.0) / Float64(-sqrt(2.0)));
	elseif (t <= -4.5e-158)
		tmp = t_1;
	elseif (t <= 2.9e-162)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	elseif (t <= 6.6e+44)
		tmp = t_1;
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.6e+119], N[(N[Sqrt[2.0], $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, -4.5e-158], t$95$1, If[LessEqual[t, 2.9e-162], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.6e+44], t$95$1, N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.6000000000000001e119

   1. Initial program 12.0%

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

      1. associate-*l/ 12.0%

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

   3. Simplified 12.0%

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

      1. associate-/r/ 12.0%

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

      2. fma-neg 12.0%

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

      3. metadata-eval 12.0%

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

      4. sub-neg 12.0%

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

      5. fma-udef 12.0%

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

      6. +-commutative 12.0%

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

      7. fma-neg 12.0%

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

   5. Applied egg-rr 4.4%

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

   6. Taylor expanded in x around inf 11.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot {t}^{2}}}}{t}} \]
   7. Step-by-step derivation

      1. unpow2 11.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \color{blue}{\left(t \cdot t\right)}}}{t}} \]

   8. Simplified 11.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot \left(t \cdot t\right)}}}{t}} \]

   9. Taylor expanded in t around -inf 93.0%

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

       1. mul-1-neg 93.0%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   11. Simplified 93.0%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   if -4.6000000000000001e119 < t < -4.5e-158 or 2.9000000000000001e-162 < t < 6.60000000000000027e44

   1. Initial program 59.7%

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

      1. associate-*l/ 59.9%

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

   3. Simplified 59.9%

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

      1. *-commutative 59.9%

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

      2. clear-num 59.7%

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

      3. un-div-inv 59.6%

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

      4. sqrt-undiv 60.0%

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

      5. metadata-eval 60.0%

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

      6. sub-neg 60.0%

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

      7. associate-*l/ 52.4%

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

      8. sub-neg 52.4%

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

      9. metadata-eval 52.4%

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

   5. Applied egg-rr 52.4%

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

      1. +-commutative 52.4%

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

      2. *-commutative 52.4%

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

      3. +-commutative 52.4%

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

      4. +-commutative 52.4%

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

   7. Simplified 52.4%

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

   8. Taylor expanded in x around inf 89.2%

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

      1. unpow2 89.2%

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

      2. +-commutative 89.2%

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

      3. distribute-lft-out 89.2%

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

      4. unpow2 89.2%

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

      5. unpow2 89.2%

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

      6. associate-*l/ 89.2%

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

      7. distribute-rgt-out 89.2%

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

      8. associate-*r/ 89.2%

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

      9. mul-1-neg 89.2%

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

      10. +-commutative 89.2%

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

      11. unpow2 89.2%

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

      12. unpow2 89.2%

          \[\leadsto \frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) - \frac{-\left(2 \cdot \left(t \cdot t\right) + \color{blue}{\ell \cdot \ell}\right)}{x}}{2}}} \]

      13. fma-udef 89.2%

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

   10. Simplified 89.2%

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

   if -4.5e-158 < t < 2.9000000000000001e-162

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in x around -inf 48.4%

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

      1. associate--l+ 48.4%

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

      2. unpow2 48.4%

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

      3. sub-neg 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in l around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*r/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. unpow2 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 57.2%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 57.2%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 6.60000000000000027e44 < t

   1. Initial program 39.6%

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

      1. associate-*l/ 39.8%

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

   3. Simplified 39.8%

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

   4. Taylor expanded in t around inf 95.3%

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

      1. associate-*l* 95.4%

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

      2. sub-neg 95.4%

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

      3. metadata-eval 95.4%

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

      4. +-commutative 95.4%

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

      5. +-commutative 95.4%

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

   6. Simplified 95.4%

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

      1. add-cbrt-cube 96.7%

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

      2. sqrt-unprod 96.7%

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

      3. +-commutative 96.7%

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

      4. sqrt-unprod 96.7%

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

      5. +-commutative 96.7%

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

   8. Applied egg-rr 96.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 96.7%

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

      2. cube-unmult 96.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 96.7%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 96.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 96.7%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 95.4%

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

       3. sqrt-unprod 96.8%

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

       4. sub-neg 96.8%

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

       5. metadata-eval 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 8: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-118}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\left(2 + \frac{2}{x}\right) + \left(\frac{2}{x} + \frac{4}{x \cdot x}\right)}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5.5e-118)
   (*
    t
    (/
     (sqrt 2.0)
     (* t (- (sqrt (+ (+ 2.0 (/ 2.0 x)) (+ (/ 2.0 x) (/ 4.0 (* x x)))))))))
   (if (<= t 1.9e-161)
     (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
     (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.5e-118) {
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 + (2.0 / x)) + ((2.0 / x) + (4.0 / (x * x)))))));
	} else if (t <= 1.9e-161) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.5d-118)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt(((2.0d0 + (2.0d0 / x)) + ((2.0d0 / x) + (4.0d0 / (x * x)))))))
    else if (t <= 1.9d-161) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.5e-118) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt(((2.0 + (2.0 / x)) + ((2.0 / x) + (4.0 / (x * x)))))));
	} else if (t <= 1.9e-161) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5.5e-118:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt(((2.0 + (2.0 / x)) + ((2.0 / x) + (4.0 / (x * x)))))))
	elif t <= 1.9e-161:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5.5e-118)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(Float64(2.0 + Float64(2.0 / x)) + Float64(Float64(2.0 / x) + Float64(4.0 / Float64(x * x)))))))));
	elseif (t <= 1.9e-161)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5.5e-118)
		tmp = t * (sqrt(2.0) / (t * -sqrt(((2.0 + (2.0 / x)) + ((2.0 / x) + (4.0 / (x * x)))))));
	elseif (t <= 1.9e-161)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5.5e-118], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / x), $MachinePrecision] + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-161], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.5000000000000003e-118

   1. Initial program 40.8%

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

      1. associate-*l/ 41.0%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in x around -inf 50.3%

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

      1. associate--l+ 50.3%

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

      2. unpow2 50.3%

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

      3. sub-neg 50.3%

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

   6. Simplified 50.3%

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

   7. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. *-commutative 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. distribute-lft-in 87.2%

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

      5. metadata-eval 87.2%

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

      6. associate-*r/ 87.2%

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

      7. metadata-eval 87.2%

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

      8. associate-*r/ 87.2%

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

      9. metadata-eval 87.2%

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

      10. +-commutative 87.2%

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

      11. associate-*r/ 87.2%

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

      12. metadata-eval 87.2%

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

      13. unpow2 87.2%

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

   9. Simplified 87.2%

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

   if -5.5000000000000003e-118 < t < 1.9000000000000001e-161

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in x around -inf 55.9%

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

      1. associate--l+ 55.9%

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

      2. unpow2 55.9%

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

      3. sub-neg 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in l around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. associate-*r/ 50.7%

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

      3. metadata-eval 50.7%

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

      4. unpow2 50.7%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 50.7%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 50.7%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 50.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 1.9000000000000001e-161 < t

   1. Initial program 45.9%

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

      1. associate-*l/ 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in t around inf 84.1%

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

      1. associate-*l* 84.1%

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

      2. sub-neg 84.1%

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

      3. metadata-eval 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

   6. Simplified 84.1%

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

      1. add-cbrt-cube 85.3%

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

      2. sqrt-unprod 85.3%

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

      3. +-commutative 85.3%

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

      4. sqrt-unprod 85.4%

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

      5. +-commutative 85.4%

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

   8. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 85.4%

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

      2. cube-unmult 85.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 85.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 85.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 85.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 85.4%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 84.1%

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

       3. sqrt-unprod 85.4%

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

       4. sub-neg 85.4%

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

       5. metadata-eval 85.4%

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

   12. Applied egg-rr 85.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-118}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{\left(2 + \frac{2}{x}\right) + \left(\frac{2}{x} + \frac{4}{x \cdot x}\right)}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-161}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 9: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.75e-118)
   (/ (sqrt 2.0) (- (sqrt 2.0)))
   (if (<= t 3.4e-163)
     (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 2.0 x) (/ 2.0 (* x x)))))))
     (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.75e-118) {
		tmp = sqrt(2.0) / -sqrt(2.0);
	} else if (t <= 3.4e-163) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.75d-118)) then
        tmp = sqrt(2.0d0) / -sqrt(2.0d0)
    else if (t <= 3.4d-163) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((2.0d0 / x) + (2.0d0 / (x * x))))))
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.75e-118) {
		tmp = Math.sqrt(2.0) / -Math.sqrt(2.0);
	} else if (t <= 3.4e-163) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((2.0 / x) + (2.0 / (x * x))))));
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.75e-118:
		tmp = math.sqrt(2.0) / -math.sqrt(2.0)
	elif t <= 3.4e-163:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((2.0 / x) + (2.0 / (x * x))))))
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.75e-118)
		tmp = Float64(sqrt(2.0) / Float64(-sqrt(2.0)));
	elseif (t <= 3.4e-163)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(2.0 / x) + Float64(2.0 / Float64(x * x)))))));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.75e-118)
		tmp = sqrt(2.0) / -sqrt(2.0);
	elseif (t <= 3.4e-163)
		tmp = t * (sqrt(2.0) / (l * sqrt(((2.0 / x) + (2.0 / (x * x))))));
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.75e-118], N[(N[Sqrt[2.0], $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 3.4e-163], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.75e-118

   1. Initial program 40.8%

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

      1. associate-*l/ 41.0%

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

   3. Simplified 41.0%

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

      1. associate-/r/ 40.9%

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

      2. fma-neg 40.9%

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

      3. metadata-eval 40.9%

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

      4. sub-neg 40.9%

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

      5. fma-udef 40.9%

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

      6. +-commutative 40.9%

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

      7. fma-neg 40.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in x around inf 46.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot {t}^{2}}}}{t}} \]
   7. Step-by-step derivation

      1. unpow2 46.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \color{blue}{\left(t \cdot t\right)}}}{t}} \]

   8. Simplified 46.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot \left(t \cdot t\right)}}}{t}} \]

   9. Taylor expanded in t around -inf 86.8%

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

       1. mul-1-neg 86.8%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   11. Simplified 86.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   if -1.75e-118 < t < 3.40000000000000014e-163

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in x around -inf 55.9%

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

      1. associate--l+ 55.9%

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

      2. unpow2 55.9%

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

      3. sub-neg 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in l around inf 50.7%

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

      1. *-commutative 50.7%

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

      2. associate-*r/ 50.7%

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

      3. metadata-eval 50.7%

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

      4. unpow2 50.7%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{\color{blue}{x \cdot x}} + 2 \cdot \frac{1}{x}}} \cdot t \]

      5. associate-*r/ 50.7%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \color{blue}{\frac{2 \cdot 1}{x}}}} \cdot t \]

      6. metadata-eval 50.7%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{\color{blue}{2}}{x}}} \cdot t \]

   9. Simplified 50.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}} \cdot t \]

   if 3.40000000000000014e-163 < t

   1. Initial program 45.9%

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

      1. associate-*l/ 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in t around inf 84.1%

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

      1. associate-*l* 84.1%

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

      2. sub-neg 84.1%

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

      3. metadata-eval 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

   6. Simplified 84.1%

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

      1. add-cbrt-cube 85.3%

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

      2. sqrt-unprod 85.3%

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

      3. +-commutative 85.3%

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

      4. sqrt-unprod 85.4%

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

      5. +-commutative 85.4%

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

   8. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 85.4%

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

      2. cube-unmult 85.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 85.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 85.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 85.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 85.4%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 84.1%

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

       3. sqrt-unprod 85.4%

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

       4. sub-neg 85.4%

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

       5. metadata-eval 85.4%

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

   12. Applied egg-rr 85.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x} + \frac{2}{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 10: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2.7e-118)
   (/ (sqrt 2.0) (- (sqrt 2.0)))
   (if (<= t 7.5e-155)
     (* t (/ (sqrt 2.0) (* l (sqrt (/ 2.0 x)))))
     (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.7e-118) {
		tmp = sqrt(2.0) / -sqrt(2.0);
	} else if (t <= 7.5e-155) {
		tmp = t * (sqrt(2.0) / (l * sqrt((2.0 / x))));
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.7d-118)) then
        tmp = sqrt(2.0d0) / -sqrt(2.0d0)
    else if (t <= 7.5d-155) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt((2.0d0 / x))))
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -2.7e-118) {
		tmp = Math.sqrt(2.0) / -Math.sqrt(2.0);
	} else if (t <= 7.5e-155) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt((2.0 / x))));
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2.7e-118:
		tmp = math.sqrt(2.0) / -math.sqrt(2.0)
	elif t <= 7.5e-155:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt((2.0 / x))))
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2.7e-118)
		tmp = Float64(sqrt(2.0) / Float64(-sqrt(2.0)));
	elseif (t <= 7.5e-155)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(2.0 / x)))));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -2.7e-118)
		tmp = sqrt(2.0) / -sqrt(2.0);
	elseif (t <= 7.5e-155)
		tmp = t * (sqrt(2.0) / (l * sqrt((2.0 / x))));
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2.7e-118], N[(N[Sqrt[2.0], $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 7.5e-155], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.69999999999999994e-118

   1. Initial program 40.8%

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

      1. associate-*l/ 41.0%

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

   3. Simplified 41.0%

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

      1. associate-/r/ 40.9%

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

      2. fma-neg 40.9%

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

      3. metadata-eval 40.9%

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

      4. sub-neg 40.9%

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

      5. fma-udef 40.9%

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

      6. +-commutative 40.9%

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

      7. fma-neg 40.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in x around inf 46.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot {t}^{2}}}}{t}} \]
   7. Step-by-step derivation

      1. unpow2 46.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \color{blue}{\left(t \cdot t\right)}}}{t}} \]

   8. Simplified 46.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot \left(t \cdot t\right)}}}{t}} \]

   9. Taylor expanded in t around -inf 86.8%

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

       1. mul-1-neg 86.8%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   11. Simplified 86.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   if -2.69999999999999994e-118 < t < 7.5000000000000006e-155

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in l around inf 6.1%

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

      1. *-commutative 6.1%

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

      2. associate--l+ 6.2%

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

      3. sub-neg 6.2%

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

      4. metadata-eval 6.2%

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

      5. +-commutative 6.2%

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

      6. sub-neg 6.2%

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

      7. metadata-eval 6.2%

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

      8. +-commutative 6.2%

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

   6. Simplified 6.2%

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

   7. Taylor expanded in x around inf 49.5%

      \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \cdot t \]

   if 7.5000000000000006e-155 < t

   1. Initial program 45.9%

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

      1. associate-*l/ 46.1%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in t around inf 84.1%

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

      1. associate-*l* 84.1%

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

      2. sub-neg 84.1%

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

      3. metadata-eval 84.1%

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

      4. +-commutative 84.1%

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

      5. +-commutative 84.1%

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

   6. Simplified 84.1%

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

      1. add-cbrt-cube 85.3%

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

      2. sqrt-unprod 85.3%

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

      3. +-commutative 85.3%

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

      4. sqrt-unprod 85.4%

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

      5. +-commutative 85.4%

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

   8. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 85.4%

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

      2. cube-unmult 85.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 85.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 85.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 85.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 85.4%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 84.1%

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

       3. sqrt-unprod 85.4%

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

       4. sub-neg 85.4%

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

       5. metadata-eval 85.4%

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

   12. Applied egg-rr 85.4%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-155}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 11: 76.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310)
   (* (sqrt 2.0) (- (sqrt 0.5)))
   (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = sqrt(2.0) * -sqrt(0.5);
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = sqrt(2.0d0) * -sqrt(0.5d0)
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = Math.sqrt(2.0) * -Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = math.sqrt(2.0) * -math.sqrt(0.5)
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(0.5)));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = sqrt(2.0) * -sqrt(0.5);
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

   4. Taylor expanded in t around inf 1.7%

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

      1. associate-*l* 1.7%

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

      2. sub-neg 1.7%

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

      3. metadata-eval 1.7%

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

      4. +-commutative 1.7%

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

      5. +-commutative 1.7%

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

   6. Simplified 1.7%

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

   7. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow2 0.0%

         \[\leadsto \sqrt{2} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \sqrt{0.5}\right) \]

      2. rem-square-sqrt 74.1%

         \[\leadsto \sqrt{2} \cdot \left(\color{blue}{-1} \cdot \sqrt{0.5}\right) \]

      3. mul-1-neg 74.1%

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

   9. Simplified 74.1%

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

   if -4.999999999999985e-310 < t

   1. Initial program 35.2%

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

      1. associate-*l/ 35.4%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in t around inf 69.0%

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

      1. associate-*l* 69.1%

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

      2. sub-neg 69.1%

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

      3. metadata-eval 69.1%

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

      4. +-commutative 69.1%

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

      5. +-commutative 69.1%

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

   6. Simplified 69.1%

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

      1. add-cbrt-cube 70.0%

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

      2. sqrt-unprod 70.0%

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

      3. +-commutative 70.0%

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

      4. sqrt-unprod 70.0%

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

      5. +-commutative 70.0%

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

   8. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 70.0%

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

      2. cube-unmult 70.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 70.0%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 70.0%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 70.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 70.0%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 69.1%

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

       3. sqrt-unprod 70.1%

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

       4. sub-neg 70.1%

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

       5. metadata-eval 70.1%

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

   12. Applied egg-rr 70.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 12: 76.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310)
   (/ (sqrt 2.0) (- (sqrt 2.0)))
   (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = sqrt(2.0) / -sqrt(2.0);
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5d-310)) then
        tmp = sqrt(2.0d0) / -sqrt(2.0d0)
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = Math.sqrt(2.0) / -Math.sqrt(2.0);
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = math.sqrt(2.0) / -math.sqrt(2.0)
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(sqrt(2.0) / Float64(-sqrt(2.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = sqrt(2.0) / -sqrt(2.0);
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5e-310], N[(N[Sqrt[2.0], $MachinePrecision] / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 35.8%

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

      1. associate-*l/ 35.9%

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

   3. Simplified 35.9%

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

      1. associate-/r/ 35.8%

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

      2. fma-neg 35.8%

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

      3. metadata-eval 35.8%

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

      4. sub-neg 35.8%

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

      5. fma-udef 35.8%

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

      6. +-commutative 35.8%

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

      7. fma-neg 35.8%

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

   5. Applied egg-rr 27.5%

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

   6. Taylor expanded in x around inf 41.5%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot {t}^{2}}}}{t}} \]
   7. Step-by-step derivation

      1. unpow2 41.5%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{2 \cdot \color{blue}{\left(t \cdot t\right)}}}{t}} \]

   8. Simplified 41.5%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{2 \cdot \left(t \cdot t\right)}}}{t}} \]

   9. Taylor expanded in t around -inf 75.2%

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

       1. mul-1-neg 75.2%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   11. Simplified 75.2%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2}}} \]

   if -4.999999999999985e-310 < t

   1. Initial program 35.2%

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

      1. associate-*l/ 35.4%

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

   3. Simplified 35.4%

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

   4. Taylor expanded in t around inf 69.0%

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

      1. associate-*l* 69.1%

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

      2. sub-neg 69.1%

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

      3. metadata-eval 69.1%

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

      4. +-commutative 69.1%

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

      5. +-commutative 69.1%

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

   6. Simplified 69.1%

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

      1. add-cbrt-cube 70.0%

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

      2. sqrt-unprod 70.0%

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

      3. +-commutative 70.0%

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

      4. sqrt-unprod 70.0%

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

      5. +-commutative 70.0%

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

   8. Applied egg-rr 70.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 70.0%

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

      2. cube-unmult 70.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 70.0%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 70.0%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 70.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 70.0%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 69.1%

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

       3. sqrt-unprod 70.1%

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

       4. sub-neg 70.1%

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

       5. metadata-eval 70.1%

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

   12. Applied egg-rr 70.1%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{-\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 13: 57.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-48}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}{2}}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\left(t \cdot t\right) \cdot \left(2 + 2 \cdot x\right)}{x + -1} - \ell \cdot \ell}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -7e-48)
   (/ t (sqrt (/ (* 2.0 (/ (+ x 1.0) (/ (+ x -1.0) (* t t)))) 2.0)))
   (if (<= t -4.5e-158)
     (/
      t
      (sqrt (/ (- (/ (* (* t t) (+ 2.0 (* 2.0 x))) (+ x -1.0)) (* l l)) 2.0)))
     (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -7e-48) {
		tmp = t / sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0));
	} else if (t <= -4.5e-158) {
		tmp = t / sqrt((((((t * t) * (2.0 + (2.0 * x))) / (x + -1.0)) - (l * l)) / 2.0));
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-7d-48)) then
        tmp = t / sqrt(((2.0d0 * ((x + 1.0d0) / ((x + (-1.0d0)) / (t * t)))) / 2.0d0))
    else if (t <= (-4.5d-158)) then
        tmp = t / sqrt((((((t * t) * (2.0d0 + (2.0d0 * x))) / (x + (-1.0d0))) - (l * l)) / 2.0d0))
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -7e-48) {
		tmp = t / Math.sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0));
	} else if (t <= -4.5e-158) {
		tmp = t / Math.sqrt((((((t * t) * (2.0 + (2.0 * x))) / (x + -1.0)) - (l * l)) / 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -7e-48:
		tmp = t / math.sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0))
	elif t <= -4.5e-158:
		tmp = t / math.sqrt((((((t * t) * (2.0 + (2.0 * x))) / (x + -1.0)) - (l * l)) / 2.0))
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -7e-48)
		tmp = Float64(t / sqrt(Float64(Float64(2.0 * Float64(Float64(x + 1.0) / Float64(Float64(x + -1.0) / Float64(t * t)))) / 2.0)));
	elseif (t <= -4.5e-158)
		tmp = Float64(t / sqrt(Float64(Float64(Float64(Float64(Float64(t * t) * Float64(2.0 + Float64(2.0 * x))) / Float64(x + -1.0)) - Float64(l * l)) / 2.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -7e-48)
		tmp = t / sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0));
	elseif (t <= -4.5e-158)
		tmp = t / sqrt((((((t * t) * (2.0 + (2.0 * x))) / (x + -1.0)) - (l * l)) / 2.0));
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -7e-48], N[(t / N[Sqrt[N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.5e-158], N[(t / N[Sqrt[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.99999999999999982e-48

   1. Initial program 35.6%

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

      1. associate-*l/ 35.7%

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

   3. Simplified 35.7%

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

      1. *-commutative 35.7%

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

      2. clear-num 35.6%

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

      3. un-div-inv 35.6%

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

      4. sqrt-undiv 35.8%

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

      5. metadata-eval 35.8%

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

      6. sub-neg 35.8%

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

      7. associate-*l/ 23.7%

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

      8. sub-neg 23.7%

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

      9. metadata-eval 23.7%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{-1}} - \ell \cdot \ell}{2}}} \]

   5. Applied egg-rr 23.7%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative 23.7%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\left(1 + x\right)} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}} \]

      2. *-commutative 23.7%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(1 + x\right)}}{x + -1} - \ell \cdot \ell}{2}}} \]

      3. +-commutative 23.7%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \color{blue}{\left(x + 1\right)}}{x + -1} - \ell \cdot \ell}{2}}} \]

      4. +-commutative 23.7%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}} - \ell \cdot \ell}{2}}} \]

   7. Simplified 23.7%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(x + 1\right)}{-1 + x} - \ell \cdot \ell}{2}}}} \]

   8. Taylor expanded in t around inf 29.4%

      \[\leadsto \frac{t}{\sqrt{\frac{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 42.6%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}{2}}} \]

      2. +-commutative 42.6%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}{2}}} \]

      3. sub-neg 42.6%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}{2}}} \]

      4. metadata-eval 42.6%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}{2}}} \]

      5. unpow2 42.6%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + -1}{\color{blue}{t \cdot t}}}}{2}}} \]

   10. Simplified 42.6%

       \[\leadsto \frac{t}{\sqrt{\frac{\color{blue}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}{2}}} \]

   if -6.99999999999999982e-48 < t < -4.5e-158

   1. Initial program 55.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 56.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. *-commutative 56.0%

         \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

      2. clear-num 55.8%

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

      3. un-div-inv 55.8%

         \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

      4. sqrt-undiv 56.2%

         \[\leadsto \frac{t}{\color{blue}{\sqrt{\frac{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}}} \]

      5. metadata-eval 56.2%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{x + 1}{x + \color{blue}{\left(-1\right)}} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}} \]

      6. sub-neg 56.2%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{x + 1}{\color{blue}{x - 1}} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}} \]

      7. associate-*l/ 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\color{blue}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x - 1}} - \ell \cdot \ell}{2}}} \]

      8. sub-neg 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x + \left(-1\right)}} - \ell \cdot \ell}{2}}} \]

      9. metadata-eval 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{-1}} - \ell \cdot \ell}{2}}} \]

   5. Applied egg-rr 56.4%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\left(1 + x\right)} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}} \]

      2. *-commutative 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(1 + x\right)}}{x + -1} - \ell \cdot \ell}{2}}} \]

      3. +-commutative 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \color{blue}{\left(x + 1\right)}}{x + -1} - \ell \cdot \ell}{2}}} \]

      4. +-commutative 56.4%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}} - \ell \cdot \ell}{2}}} \]

   7. Simplified 56.4%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(x + 1\right)}{-1 + x} - \ell \cdot \ell}{2}}}} \]

   8. Taylor expanded in t around inf 52.0%

      \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{2 \cdot \left(\left(1 + x\right) \cdot {t}^{2}\right)}}{-1 + x} - \ell \cdot \ell}{2}}} \]
   9. Step-by-step derivation

      1. associate-*r* 52.0%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\left(2 \cdot \left(1 + x\right)\right) \cdot {t}^{2}}}{-1 + x} - \ell \cdot \ell}{2}}} \]

      2. distribute-lft-in 52.0%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\left(2 \cdot 1 + 2 \cdot x\right)} \cdot {t}^{2}}{-1 + x} - \ell \cdot \ell}{2}}} \]

      3. metadata-eval 52.0%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(\color{blue}{2} + 2 \cdot x\right) \cdot {t}^{2}}{-1 + x} - \ell \cdot \ell}{2}}} \]

      4. unpow2 52.0%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(2 + 2 \cdot x\right) \cdot \color{blue}{\left(t \cdot t\right)}}{-1 + x} - \ell \cdot \ell}{2}}} \]

   10. Simplified 52.0%

       \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\left(2 + 2 \cdot x\right) \cdot \left(t \cdot t\right)}}{-1 + x} - \ell \cdot \ell}{2}}} \]

   if -4.5e-158 < t

   1. Initial program 31.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 32.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 62.5%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   5. Step-by-step derivation

      1. associate-*l* 62.5%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]

      2. sub-neg 62.5%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]

      3. metadata-eval 62.5%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]

      4. +-commutative 62.5%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]

      5. +-commutative 62.5%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   6. Simplified 62.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 63.4%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

      2. sqrt-unprod 63.4%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{0.5 \cdot \frac{-1 + x}{x + 1}}}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

      3. +-commutative 63.4%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x + -1}}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

      4. sqrt-unprod 63.4%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{0.5 \cdot \frac{-1 + x}{x + 1}}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

      5. +-commutative 63.4%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x + -1}}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

   8. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 63.4%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]

      2. cube-unmult 63.4%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 63.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 63.4%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 63.4%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 63.4%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 62.5%

          \[\leadsto 1 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)} \]

       3. sqrt-unprod 63.4%

          \[\leadsto 1 \cdot \color{blue}{\sqrt{2 \cdot \left(0.5 \cdot \frac{x - 1}{x + 1}\right)}} \]

       4. sub-neg 63.4%

          \[\leadsto 1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{x + \left(-1\right)}}{x + 1}\right)} \]

       5. metadata-eval 63.4%

          \[\leadsto 1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + \color{blue}{-1}}{x + 1}\right)} \]

   12. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-48}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}{2}}}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\frac{\left(t \cdot t\right) \cdot \left(2 + 2 \cdot x\right)}{x + -1} - \ell \cdot \ell}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 14: 55.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5.2e-160)
   (/ t (sqrt (/ (* 2.0 (/ (+ x 1.0) (/ (+ x -1.0) (* t t)))) 2.0)))
   (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.2e-160) {
		tmp = t / sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0));
	} else {
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.2d-160)) then
        tmp = t / sqrt(((2.0d0 * ((x + 1.0d0) / ((x + (-1.0d0)) / (t * t)))) / 2.0d0))
    else
        tmp = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -5.2e-160) {
		tmp = t / Math.sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0));
	} else {
		tmp = Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5.2e-160:
		tmp = t / math.sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0))
	else:
		tmp = math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5.2e-160)
		tmp = Float64(t / sqrt(Float64(Float64(2.0 * Float64(Float64(x + 1.0) / Float64(Float64(x + -1.0) / Float64(t * t)))) / 2.0)));
	else
		tmp = sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5.2e-160)
		tmp = t / sqrt(((2.0 * ((x + 1.0) / ((x + -1.0) / (t * t)))) / 2.0));
	else
		tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -5.2e-160], N[(t / N[Sqrt[N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[(N[(x + -1.0), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.20000000000000007e-160

   1. Initial program 39.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 39.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 39.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. *-commutative 39.8%

         \[\leadsto \color{blue}{t \cdot \frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

      2. clear-num 39.7%

         \[\leadsto t \cdot \color{blue}{\frac{1}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

      3. un-div-inv 39.7%

         \[\leadsto \color{blue}{\frac{t}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{\sqrt{2}}}} \]

      4. sqrt-undiv 39.9%

         \[\leadsto \frac{t}{\color{blue}{\sqrt{\frac{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}}} \]

      5. metadata-eval 39.9%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{x + 1}{x + \color{blue}{\left(-1\right)}} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}} \]

      6. sub-neg 39.9%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{x + 1}{\color{blue}{x - 1}} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}{2}}} \]

      7. associate-*l/ 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\color{blue}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x - 1}} - \ell \cdot \ell}{2}}} \]

      8. sub-neg 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x + \left(-1\right)}} - \ell \cdot \ell}{2}}} \]

      9. metadata-eval 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{-1}} - \ell \cdot \ell}{2}}} \]

   5. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}}} \]
   6. Step-by-step derivation

      1. +-commutative 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\left(1 + x\right)} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}{2}}} \]

      2. *-commutative 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(1 + x\right)}}{x + -1} - \ell \cdot \ell}{2}}} \]

      3. +-commutative 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \color{blue}{\left(x + 1\right)}}{x + -1} - \ell \cdot \ell}{2}}} \]

      4. +-commutative 30.6%

         \[\leadsto \frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(x + 1\right)}{\color{blue}{-1 + x}} - \ell \cdot \ell}{2}}} \]

   7. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{t}{\sqrt{\frac{\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) \cdot \left(x + 1\right)}{-1 + x} - \ell \cdot \ell}{2}}}} \]

   8. Taylor expanded in t around inf 37.5%

      \[\leadsto \frac{t}{\sqrt{\frac{\color{blue}{2 \cdot \frac{\left(1 + x\right) \cdot {t}^{2}}{x - 1}}}{2}}} \]
   9. Step-by-step derivation

      1. associate-/l* 37.1%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \color{blue}{\frac{1 + x}{\frac{x - 1}{{t}^{2}}}}}{2}}} \]

      2. +-commutative 37.1%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{\color{blue}{x + 1}}{\frac{x - 1}{{t}^{2}}}}{2}}} \]

      3. sub-neg 37.1%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{{t}^{2}}}}{2}}} \]

      4. metadata-eval 37.1%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + \color{blue}{-1}}{{t}^{2}}}}{2}}} \]

      5. unpow2 37.1%

         \[\leadsto \frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + -1}{\color{blue}{t \cdot t}}}}{2}}} \]

   10. Simplified 37.1%

       \[\leadsto \frac{t}{\sqrt{\frac{\color{blue}{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}}{2}}} \]

   if -5.20000000000000007e-160 < t

   1. Initial program 32.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 32.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 32.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 62.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   5. Step-by-step derivation

      1. associate-*l* 62.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]

      2. sub-neg 62.9%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]

      3. metadata-eval 62.9%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]

      4. +-commutative 62.9%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]

      5. +-commutative 62.9%

         \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

   6. Simplified 62.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
   7. Step-by-step derivation

      1. add-cbrt-cube 63.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

      2. sqrt-unprod 63.8%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{0.5 \cdot \frac{-1 + x}{x + 1}}}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

      3. +-commutative 63.8%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x + -1}}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

      4. sqrt-unprod 63.8%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{0.5 \cdot \frac{-1 + x}{x + 1}}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

      5. +-commutative 63.8%

         \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x + -1}}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

   8. Applied egg-rr 63.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
   9. Step-by-step derivation

      1. associate-*l* 63.8%

         \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]

      2. cube-unmult 63.8%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

      3. metadata-eval 63.8%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

      4. sub-neg 63.8%

         \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

   10. Simplified 63.8%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
   11. Step-by-step derivation

       1. *-un-lft-identity 63.8%

          \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

       2. rem-cbrt-cube 62.9%

          \[\leadsto 1 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)} \]

       3. sqrt-unprod 63.8%

          \[\leadsto 1 \cdot \color{blue}{\sqrt{2 \cdot \left(0.5 \cdot \frac{x - 1}{x + 1}\right)}} \]

       4. sub-neg 63.8%

          \[\leadsto 1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{x + \left(-1\right)}}{x + 1}\right)} \]

       5. metadata-eval 63.8%

          \[\leadsto 1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + \color{blue}{-1}}{x + 1}\right)} \]

   12. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{2 \cdot \frac{x + 1}{\frac{x + -1}{t \cdot t}}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}\\ \end{array} \]

Alternative 15: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (sqrt (* 2.0 (* 0.5 (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	return sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = sqrt((2.0d0 * (0.5d0 * ((x + (-1.0d0)) / (x + 1.0d0)))))
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return Math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
}

Python

l = abs(l)
def code(x, l, t):
	return math.sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))))

Julia

l = abs(l)
function code(x, l, t)
	return sqrt(Float64(2.0 * Float64(0.5 * Float64(Float64(x + -1.0) / Float64(x + 1.0)))))
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = sqrt((2.0 * (0.5 * ((x + -1.0) / (x + 1.0)))));
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := N[Sqrt[N[(2.0 * N[(0.5 * N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 35.5%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 35.7%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around inf 35.4%

   \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
5. Step-by-step derivation

   1. associate-*l* 35.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x - 1}{1 + x}}\right)} \]

   2. sub-neg 35.4%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}}\right) \]

   3. metadata-eval 35.4%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{x + \color{blue}{-1}}{1 + x}}\right) \]

   4. +-commutative 35.4%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{\color{blue}{-1 + x}}{1 + x}}\right) \]

   5. +-commutative 35.4%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{\color{blue}{x + 1}}}\right) \]

6. Simplified 35.4%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)} \]
7. Step-by-step derivation

   1. add-cbrt-cube 35.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)}} \]

   2. sqrt-unprod 35.9%

      \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \color{blue}{\sqrt{0.5 \cdot \frac{-1 + x}{x + 1}}}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

   3. +-commutative 35.9%

      \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x + -1}}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

   4. sqrt-unprod 35.9%

      \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{0.5 \cdot \frac{-1 + x}{x + 1}}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

   5. +-commutative 35.9%

      \[\leadsto \sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x + -1}}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \left(\sqrt{0.5} \cdot \sqrt{\frac{-1 + x}{x + 1}}\right)\right)} \]

8. Applied egg-rr 35.9%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}} \]
9. Step-by-step derivation

   1. associate-*l* 35.9%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right) \cdot \left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)\right)}} \]

   2. cube-unmult 35.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + -1}{x + 1}}\right)}^{3}}} \]

   3. metadata-eval 35.9%

      \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x + \color{blue}{\left(-1\right)}}{x + 1}}\right)}^{3}} \]

   4. sub-neg 35.9%

      \[\leadsto \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{\color{blue}{x - 1}}{x + 1}}\right)}^{3}} \]

10. Simplified 35.9%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 35.9%

       \[\leadsto \color{blue}{1 \cdot \sqrt[3]{{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)}^{3}}} \]

    2. rem-cbrt-cube 35.4%

       \[\leadsto 1 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{0.5 \cdot \frac{x - 1}{x + 1}}\right)} \]

    3. sqrt-unprod 35.9%

       \[\leadsto 1 \cdot \color{blue}{\sqrt{2 \cdot \left(0.5 \cdot \frac{x - 1}{x + 1}\right)}} \]

    4. sub-neg 35.9%

       \[\leadsto 1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{\color{blue}{x + \left(-1\right)}}{x + 1}\right)} \]

    5. metadata-eval 35.9%

       \[\leadsto 1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + \color{blue}{-1}}{x + 1}\right)} \]

12. Applied egg-rr 35.9%

    \[\leadsto \color{blue}{1 \cdot \sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)}} \]

13. Final simplification 35.9%

    \[\leadsto \sqrt{2 \cdot \left(0.5 \cdot \frac{x + -1}{x + 1}\right)} \]

Alternative 16: 39.1% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return 1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = 1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return 1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return 1.0

Julia

l = abs(l)
function code(x, l, t)
	return 1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = 1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := 1.0

Derivation

1. Initial program 35.5%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 35.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 35.7%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in x around inf 35.3%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot t}} \cdot t \]

5. Taylor expanded in t around 0 35.5%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 35.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))