Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 82.9%

Time: 24.9s

Alternatives: 17

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: 33.4% 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: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{2}{x} + \left(2 + \frac{2}{x}\right)\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := \frac{\ell \cdot \ell}{x}\\ t_4 := t \cdot \frac{\sqrt{2}}{\sqrt{t_3 + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-262}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(-0.5 \cdot \frac{t_3 + t_3}{t}\right) \cdot \sqrt{\frac{1}{t_1}} - t \cdot \sqrt{t_1}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.00192:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_3 (/ (* l l) x))
        (t_4
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             t_3
             (+
              (* 2.0 (+ (* t t) (/ (* t t) x)))
              (/ (fma 2.0 (* t t) (* l l)) x))))))))
   (if (<= t -1.45e-31)
     (- t_2)
     (if (<= t -1.7e-262)
       t_4
       (if (<= t -1.45e-305)
         (*
          (sqrt 2.0)
          (/
           t
           (-
            (* (* -0.5 (/ (+ t_3 t_3) t)) (sqrt (/ 1.0 t_1)))
            (* t (sqrt t_1)))))
         (if (<= t 8.5e-193)
           (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x)))))
           (if (<= t 1.35e-155) 1.0 (if (<= t 0.00192) t_4 t_2))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (2.0 / x) + (2.0 + (2.0 / x));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = (l * l) / x;
	double t_4 = t * (sqrt(2.0) / sqrt((t_3 + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x)))));
	double tmp;
	if (t <= -1.45e-31) {
		tmp = -t_2;
	} else if (t <= -1.7e-262) {
		tmp = t_4;
	} else if (t <= -1.45e-305) {
		tmp = sqrt(2.0) * (t / (((-0.5 * ((t_3 + t_3) / t)) * sqrt((1.0 / t_1))) - (t * sqrt(t_1))));
	} else if (t <= 8.5e-193) {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.00192) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x)))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_3 = Float64(Float64(l * l) / x)
	t_4 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_3 + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))))
	tmp = 0.0
	if (t <= -1.45e-31)
		tmp = Float64(-t_2);
	elseif (t <= -1.7e-262)
		tmp = t_4;
	elseif (t <= -1.45e-305)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(Float64(-0.5 * Float64(Float64(t_3 + t_3) / t)) * sqrt(Float64(1.0 / t_1))) - Float64(t * sqrt(t_1)))));
	elseif (t <= 8.5e-193)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.00192)
		tmp = t_4;
	else
		tmp = 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[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.45e-31], (-t$95$2), If[LessEqual[t, -1.7e-262], t$95$4, If[LessEqual[t, -1.45e-305], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[(-0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-193], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-155], 1.0, If[LessEqual[t, 0.00192], t$95$4, t$95$2]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.45e-31

   1. Initial program 42.5%

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

      1. associate-*r/ 42.4%

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

      2. fma-neg 42.4%

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

      3. remove-double-neg 42.4%

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

      4. fma-neg 42.4%

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

      5. sub-neg 42.4%

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

      6. metadata-eval 42.4%

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

      7. remove-double-neg 42.4%

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

      8. +-commutative 42.4%

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

      9. fma-def 42.4%

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

   3. Simplified 42.4%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 92.5%

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

      3. metadata-eval 92.5%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 92.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.45e-31 < t < -1.69999999999999995e-262 or 1.34999999999999991e-155 < t < 0.00192000000000000005

   1. Initial program 43.5%

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

      1. associate-*l/ 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in x around inf 77.5%

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

      1. associate--l+ 77.5%

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

      2. unpow2 77.5%

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

      3. distribute-lft-out 77.5%

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

      4. unpow2 77.5%

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

      5. unpow2 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unpow2 77.5%

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

      9. +-commutative 77.5%

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

      10. unpow2 77.5%

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

      11. fma-udef 77.5%

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

   6. Simplified 77.5%

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

   if -1.69999999999999995e-262 < t < -1.44999999999999994e-305

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. remove-double-neg 2.9%

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

      4. fma-neg 2.9%

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

      5. sub-neg 2.9%

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

      6. metadata-eval 2.9%

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

      7. remove-double-neg 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-def 2.9%

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

   3. Simplified 2.9%

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

   4. Taylor expanded in x around inf 12.7%

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

      1. associate--l+ 12.7%

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

      2. unpow2 12.7%

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

      3. distribute-lft-out 12.7%

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

      4. unpow2 12.7%

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

      5. unpow2 12.7%

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

      6. associate-*r/ 12.7%

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

      7. mul-1-neg 12.7%

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

      8. unpow2 12.7%

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

      9. +-commutative 12.7%

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

      10. unpow2 12.7%

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

      11. fma-udef 12.7%

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

   6. Simplified 12.7%

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

   7. Taylor expanded in t around -inf 76.6%

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

      1. mul-1-neg 76.6%

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

      2. *-commutative 76.6%

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

      3. unsub-neg 76.6%

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

   9. Simplified 76.6%

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

   if -1.44999999999999994e-305 < t < 8.50000000000000004e-193

   1. Initial program 4.3%

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

      1. associate-*r/ 4.3%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.3%

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

      5. sub-neg 4.3%

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

      6. metadata-eval 4.3%

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

      7. remove-double-neg 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-def 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in x around inf 64.7%

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

      1. associate--l+ 64.6%

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

      2. unpow2 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. unpow2 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r/ 64.6%

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

      7. mul-1-neg 64.6%

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

      8. unpow2 64.6%

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

      9. +-commutative 64.6%

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

      10. unpow2 64.6%

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

      11. fma-udef 64.6%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in l around inf 55.5%

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

   if 8.50000000000000004e-193 < t < 1.34999999999999991e-155

   1. Initial program 12.8%

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

      1. associate-*r/ 12.8%

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

      2. fma-neg 12.8%

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

      3. remove-double-neg 12.8%

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

      4. fma-neg 12.8%

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

      5. sub-neg 12.8%

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

      6. metadata-eval 12.8%

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

      7. remove-double-neg 12.8%

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

      8. +-commutative 12.8%

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

      9. fma-def 12.8%

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

   3. Simplified 12.8%

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

   5. Applied egg-rr 47.8%

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

   6. Taylor expanded in t around -inf 1.6%

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

      1. mul-1-neg 1.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.2%

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

   11. Simplified 79.2%

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

   if 0.00192000000000000005 < t

   1. Initial program 39.5%

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

      1. associate-*r/ 39.4%

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

      2. fma-neg 39.4%

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

      3. remove-double-neg 39.4%

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

      4. fma-neg 39.4%

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

      5. sub-neg 39.4%

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

      6. metadata-eval 39.4%

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

      7. remove-double-neg 39.4%

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

      8. +-commutative 39.4%

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

      9. fma-def 39.4%

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

   3. Simplified 39.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in t around inf 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-262}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(-0.5 \cdot \frac{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}{t}\right) \cdot \sqrt{\frac{1}{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}} - t \cdot \sqrt{\frac{2}{x} + \left(2 + \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.00192:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2} \cdot t\\ t_2 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_3 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{t_2 + t_2}{x \cdot x}\right) + \left(\frac{t_2}{{x}^{3}} + \frac{t_2}{x}\right)\right)}}\\ t_4 := \frac{x + 1}{x + -1}\\ t_5 := \frac{t_1}{\sqrt{t_4 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\\ \mathbf{if}\;t_5 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(t_1 \cdot \sqrt{t_4}, \ell\right)}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) t))
        (t_2 (fma 2.0 (* t t) (* l l)))
        (t_3
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+
              (fma
               2.0
               (/ (* t t) (pow x 3.0))
               (+
                (+ (* 2.0 (+ (* t t) (/ (* t t) x))) (/ (* l l) (pow x 3.0)))
                (/ (+ t_2 t_2) (* x x))))
              (+ (/ t_2 (pow x 3.0)) (/ t_2 x))))))))
        (t_4 (/ (+ x 1.0) (+ x -1.0)))
        (t_5 (/ t_1 (sqrt (- (* t_4 (+ (* l l) (* 2.0 (* t t)))) (* l l))))))
   (if (<= t_5 -1e-299)
     t_3
     (if (<= t_5 0.0)
       (/ t_1 (hypot (* t_1 (sqrt t_4)) l))
       (if (<= t_5 INFINITY)
         t_3
         (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * t;
	double t_2 = fma(2.0, (t * t), (l * l));
	double t_3 = t * (sqrt(2.0) / sqrt((((l * l) / x) + (fma(2.0, ((t * t) / pow(x, 3.0)), (((2.0 * ((t * t) + ((t * t) / x))) + ((l * l) / pow(x, 3.0))) + ((t_2 + t_2) / (x * x)))) + ((t_2 / pow(x, 3.0)) + (t_2 / x))))));
	double t_4 = (x + 1.0) / (x + -1.0);
	double t_5 = t_1 / sqrt(((t_4 * ((l * l) + (2.0 * (t * t)))) - (l * l)));
	double tmp;
	if (t_5 <= -1e-299) {
		tmp = t_3;
	} else if (t_5 <= 0.0) {
		tmp = t_1 / hypot((t_1 * sqrt(t_4)), l);
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * t)
	t_2 = fma(2.0, Float64(t * t), Float64(l * l))
	t_3 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(fma(2.0, Float64(Float64(t * t) / (x ^ 3.0)), Float64(Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(Float64(l * l) / (x ^ 3.0))) + Float64(Float64(t_2 + t_2) / Float64(x * x)))) + Float64(Float64(t_2 / (x ^ 3.0)) + Float64(t_2 / x)))))))
	t_4 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	t_5 = Float64(t_1 / sqrt(Float64(Float64(t_4 * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
	tmp = 0.0
	if (t_5 <= -1e-299)
		tmp = t_3;
	elseif (t_5 <= 0.0)
		tmp = Float64(t_1 / hypot(Float64(t_1 * sqrt(t_4)), l));
	elseif (t_5 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 / N[Sqrt[N[(N[(t$95$4 * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -1e-299], t$95$3, If[LessEqual[t$95$5, 0.0], N[(t$95$1 / N[Sqrt[N[(t$95$1 * N[Sqrt[t$95$4], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], t$95$3, N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -9.99999999999999992e-300 or -0.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 57.2%

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

      1. associate-*l/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around -inf 83.7%

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

   6. Simplified 83.7%

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

   if -9.99999999999999992e-300 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -0.0

   1. Initial program 3.1%

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

      1. associate-*r/ 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. +-commutative 3.1%

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

      9. fma-def 3.1%

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

   3. Simplified 3.1%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 100.0%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. fma-neg 0.0%

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

      3. remove-double-neg 0.0%

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

      4. fma-neg 0.0%

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

      5. sub-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. remove-double-neg 0.0%

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

      8. +-commutative 0.0%

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

      9. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around inf 18.2%

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

      1. associate--l+ 18.2%

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

      2. unpow2 18.2%

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

      3. distribute-lft-out 18.2%

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

      4. unpow2 18.2%

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

      5. unpow2 18.2%

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

      6. associate-*r/ 18.2%

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

      7. mul-1-neg 18.2%

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

      8. unpow2 18.2%

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

      9. +-commutative 18.2%

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

      10. unpow2 18.2%

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

      11. fma-udef 18.2%

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

   6. Simplified 18.2%

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

   7. Taylor expanded in l around inf 42.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{{x}^{3}} + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 0:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{hypot}\left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\mathsf{fma}\left(2, \frac{t \cdot t}{{x}^{3}}, \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\ell \cdot \ell}{{x}^{3}}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{{x}^{3}} + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 3: 82.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2} \cdot t\\ t_2 := \frac{x + 1}{x + -1}\\ t_3 := \frac{t_1}{\sqrt{t_2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\\ t_4 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_5 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{t_4 + t_4}{x \cdot x}\right) + \frac{t_4}{x}\right)}}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(t_1 \cdot \sqrt{t_2}, \ell\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) t))
        (t_2 (/ (+ x 1.0) (+ x -1.0)))
        (t_3 (/ t_1 (sqrt (- (* t_2 (+ (* l l) (* 2.0 (* t t)))) (* l l)))))
        (t_4 (fma 2.0 (* t t) (* l l)))
        (t_5
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+
              (+ (* 2.0 (+ (* t t) (/ (* t t) x))) (/ (+ t_4 t_4) (* x x)))
              (/ t_4 x))))))))
   (if (<= t_3 -1e-299)
     t_5
     (if (<= t_3 0.0)
       (/ t_1 (hypot (* t_1 (sqrt t_2)) l))
       (if (<= t_3 INFINITY)
         t_5
         (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * t;
	double t_2 = (x + 1.0) / (x + -1.0);
	double t_3 = t_1 / sqrt(((t_2 * ((l * l) + (2.0 * (t * t)))) - (l * l)));
	double t_4 = fma(2.0, (t * t), (l * l));
	double t_5 = t * (sqrt(2.0) / sqrt((((l * l) / x) + (((2.0 * ((t * t) + ((t * t) / x))) + ((t_4 + t_4) / (x * x))) + (t_4 / x)))));
	double tmp;
	if (t_3 <= -1e-299) {
		tmp = t_5;
	} else if (t_3 <= 0.0) {
		tmp = t_1 / hypot((t_1 * sqrt(t_2)), l);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * t)
	t_2 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	t_3 = Float64(t_1 / sqrt(Float64(Float64(t_2 * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
	t_4 = fma(2.0, Float64(t * t), Float64(l * l))
	t_5 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(Float64(t_4 + t_4) / Float64(x * x))) + Float64(t_4 / x))))))
	tmp = 0.0
	if (t_3 <= -1e-299)
		tmp = t_5;
	elseif (t_3 <= 0.0)
		tmp = Float64(t_1 / hypot(Float64(t_1 * sqrt(t_2)), l));
	elseif (t_3 <= Inf)
		tmp = t_5;
	else
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[(t$95$2 * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$4 + t$95$4), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-299], t$95$5, If[LessEqual[t$95$3, 0.0], N[(t$95$1 / N[Sqrt[N[(t$95$1 * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$5, N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -9.99999999999999992e-300 or -0.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < +inf.0

   1. Initial program 57.2%

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

      1. associate-*l/ 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around -inf 83.4%

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

      1. associate--l+ 83.4%

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

      2. unpow2 83.4%

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

      3. sub-neg 83.4%

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

   6. Simplified 83.4%

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

   if -9.99999999999999992e-300 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -0.0

   1. Initial program 3.1%

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

      1. associate-*r/ 3.1%

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

      2. fma-neg 3.1%

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

      3. remove-double-neg 3.1%

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

      4. fma-neg 3.1%

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

      5. sub-neg 3.1%

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

      6. metadata-eval 3.1%

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

      7. remove-double-neg 3.1%

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

      8. +-commutative 3.1%

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

      9. fma-def 3.1%

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

   3. Simplified 3.1%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in l around 0 100.0%

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

   if +inf.0 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 0.0%

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

      1. associate-*r/ 0.0%

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

      2. fma-neg 0.0%

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

      3. remove-double-neg 0.0%

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

      4. fma-neg 0.0%

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

      5. sub-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. remove-double-neg 0.0%

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

      8. +-commutative 0.0%

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

      9. fma-def 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around inf 18.2%

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

      1. associate--l+ 18.2%

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

      2. unpow2 18.2%

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

      3. distribute-lft-out 18.2%

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

      4. unpow2 18.2%

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

      5. unpow2 18.2%

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

      6. associate-*r/ 18.2%

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

      7. mul-1-neg 18.2%

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

      8. unpow2 18.2%

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

      9. +-commutative 18.2%

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

      10. unpow2 18.2%

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

      11. fma-udef 18.2%

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

   6. Simplified 18.2%

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

   7. Taylor expanded in l around inf 42.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 0:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{hypot}\left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq \infty:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 4: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2} \cdot t\\ t_2 := \frac{x + 1}{x + -1}\\ t_3 := \frac{t_1}{\sqrt{t_2 \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t_3 \leq 2:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(t_1 \cdot \sqrt{t_2}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (sqrt 2.0) t))
        (t_2 (/ (+ x 1.0) (+ x -1.0)))
        (t_3 (/ t_1 (sqrt (- (* t_2 (+ (* l l) (* 2.0 (* t t)))) (* l l))))))
   (if (<= t_3 -1e-299)
     (*
      t
      (/
       (sqrt 2.0)
       (sqrt
        (+
         (/ (* l l) x)
         (+
          (* 2.0 (+ (* t t) (/ (* t t) x)))
          (/ (fma 2.0 (* t t) (* l l)) x))))))
     (if (<= t_3 2.0)
       (/ t_1 (hypot (* t_1 (sqrt t_2)) l))
       (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) * t;
	double t_2 = (x + 1.0) / (x + -1.0);
	double t_3 = t_1 / sqrt(((t_2 * ((l * l) + (2.0 * (t * t)))) - (l * l)));
	double tmp;
	if (t_3 <= -1e-299) {
		tmp = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x)))));
	} else if (t_3 <= 2.0) {
		tmp = t_1 / hypot((t_1 * sqrt(t_2)), l);
	} else {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) * t)
	t_2 = Float64(Float64(x + 1.0) / Float64(x + -1.0))
	t_3 = Float64(t_1 / sqrt(Float64(Float64(t_2 * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
	tmp = 0.0
	if (t_3 <= -1e-299)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))));
	elseif (t_3 <= 2.0)
		tmp = Float64(t_1 / hypot(Float64(t_1 * sqrt(t_2)), l));
	else
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[(t$95$2 * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-299], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$1 / N[Sqrt[N[(t$95$1 * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2 + l ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < -9.99999999999999992e-300

   1. Initial program 52.0%

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

      1. associate-*l/ 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in x around inf 80.4%

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

      1. associate--l+ 80.4%

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

      2. unpow2 80.4%

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

      3. distribute-lft-out 80.4%

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

      4. unpow2 80.4%

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

      5. unpow2 80.4%

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

      6. associate-*r/ 80.4%

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

      7. mul-1-neg 80.4%

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

      8. unpow2 80.4%

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

      9. +-commutative 80.4%

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

      10. unpow2 80.4%

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

      11. fma-udef 80.4%

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

   6. Simplified 80.4%

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

   if -9.99999999999999992e-300 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l)))) < 2

   1. Initial program 49.9%

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

      1. associate-*r/ 49.8%

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

      2. fma-neg 49.8%

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

      3. remove-double-neg 49.8%

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

      4. fma-neg 49.8%

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

      5. sub-neg 49.8%

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

      6. metadata-eval 49.8%

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

      7. remove-double-neg 49.8%

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

      8. +-commutative 49.8%

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

      9. fma-def 49.8%

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

   3. Simplified 49.8%

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

   5. Applied egg-rr 98.3%

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

   6. Taylor expanded in l around 0 98.3%

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

   if 2 < (/.f64 (*.f64 (sqrt.f64 2) t) (sqrt.f64 (-.f64 (*.f64 (/.f64 (+.f64 x 1) (-.f64 x 1)) (+.f64 (*.f64 l l) (*.f64 2 (*.f64 t t)))) (*.f64 l l))))

   1. Initial program 1.2%

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

      1. associate-*r/ 1.2%

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

      2. fma-neg 1.2%

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

      3. remove-double-neg 1.2%

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

      4. fma-neg 1.2%

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

      5. sub-neg 1.2%

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

      6. metadata-eval 1.2%

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

      7. remove-double-neg 1.2%

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

      8. +-commutative 1.2%

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

      9. fma-def 1.2%

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

   3. Simplified 1.2%

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

   4. Taylor expanded in x around inf 38.7%

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

      1. associate--l+ 38.7%

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

      2. unpow2 38.7%

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

      3. distribute-lft-out 38.7%

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

      4. unpow2 38.7%

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

      5. unpow2 38.7%

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

      6. associate-*r/ 38.7%

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

      7. mul-1-neg 38.7%

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

      8. unpow2 38.7%

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

      9. +-commutative 38.7%

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

      10. unpow2 38.7%

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

      11. fma-udef 38.7%

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

   6. Simplified 38.7%

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

   7. Taylor expanded in l around inf 36.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq -1 \cdot 10^{-299}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x + -1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \leq 2:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\mathsf{hypot}\left(\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{x + 1}{x + -1}}, \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]

Alternative 5: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ t_3 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_3 + \left(t_3 + t_1\right)}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0058:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_3 + \left(t_1 + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (+ (* t t) (/ (* t t) x))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_3 (/ (* l l) x)))
   (if (<= t -1.12e-33)
     (- t_2)
     (if (<= t -2e-258)
       (* (sqrt 2.0) (/ t (sqrt (+ t_3 (+ t_3 t_1)))))
       (if (<= t -3.1e-305)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 2e-196)
           (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x)))))
           (if (<= t 1.35e-155)
             1.0
             (if (<= t 0.0058)
               (*
                (sqrt 2.0)
                (/ t (sqrt (+ t_3 (+ t_1 (/ (fma 2.0 (* t t) (* l l)) x))))))
               t_2))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 2.0 * ((t * t) + ((t * t) / x));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_3 = (l * l) / x;
	double tmp;
	if (t <= -1.12e-33) {
		tmp = -t_2;
	} else if (t <= -2e-258) {
		tmp = sqrt(2.0) * (t / sqrt((t_3 + (t_3 + t_1))));
	} else if (t <= -3.1e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 2e-196) {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.0058) {
		tmp = sqrt(2.0) * (t / sqrt((t_3 + (t_1 + (fma(2.0, (t * t), (l * l)) / x)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x)))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_3 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -1.12e-33)
		tmp = Float64(-t_2);
	elseif (t <= -2e-258)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_3 + Float64(t_3 + t_1)))));
	elseif (t <= -3.1e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 2e-196)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.0058)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_3 + Float64(t_1 + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))));
	else
		tmp = 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[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -1.12e-33], (-t$95$2), If[LessEqual[t, -2e-258], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$3 + N[(t$95$3 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.1e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 2e-196], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-155], 1.0, If[LessEqual[t, 0.0058], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$3 + N[(t$95$1 + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.11999999999999999e-33

   1. Initial program 42.5%

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

      1. associate-*r/ 42.4%

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

      2. fma-neg 42.4%

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

      3. remove-double-neg 42.4%

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

      4. fma-neg 42.4%

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

      5. sub-neg 42.4%

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

      6. metadata-eval 42.4%

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

      7. remove-double-neg 42.4%

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

      8. +-commutative 42.4%

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

      9. fma-def 42.4%

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

   3. Simplified 42.4%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 92.5%

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

      3. metadata-eval 92.5%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 92.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.11999999999999999e-33 < t < -1.99999999999999991e-258

   1. Initial program 31.1%

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

      1. associate-*r/ 31.1%

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

      2. fma-neg 31.1%

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

      3. remove-double-neg 31.1%

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

      4. fma-neg 31.1%

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

      5. sub-neg 31.1%

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

      6. metadata-eval 31.1%

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

      7. remove-double-neg 31.1%

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

      8. +-commutative 31.1%

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

      9. fma-def 31.1%

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

   3. Simplified 31.1%

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

   4. Taylor expanded in x around inf 74.4%

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

      1. associate--l+ 74.4%

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

      2. unpow2 74.4%

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

      3. distribute-lft-out 74.4%

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

      4. unpow2 74.4%

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

      5. unpow2 74.4%

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

      6. associate-*r/ 74.4%

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

      7. mul-1-neg 74.4%

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

      8. unpow2 74.4%

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

      9. +-commutative 74.4%

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

      10. unpow2 74.4%

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

      11. fma-udef 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in t around 0 74.4%

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

      1. associate-*r/ 74.4%

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

      2. mul-1-neg 74.4%

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

      3. unpow2 74.4%

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

      4. distribute-rgt-neg-in 74.4%

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

   9. Simplified 74.4%

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

   if -1.99999999999999991e-258 < t < -3.0999999999999998e-305

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. remove-double-neg 2.9%

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

      4. fma-neg 2.9%

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

      5. sub-neg 2.9%

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

      6. metadata-eval 2.9%

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

      7. remove-double-neg 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-def 2.9%

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

   3. Simplified 2.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in t around -inf 67.0%

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

      1. mul-1-neg 67.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 67.0%

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

      3. metadata-eval 67.0%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 67.0%

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

   if -3.0999999999999998e-305 < t < 2.0000000000000001e-196

   1. Initial program 4.3%

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

      1. associate-*r/ 4.3%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.3%

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

      5. sub-neg 4.3%

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

      6. metadata-eval 4.3%

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

      7. remove-double-neg 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-def 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in x around inf 64.7%

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

      1. associate--l+ 64.6%

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

      2. unpow2 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. unpow2 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r/ 64.6%

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

      7. mul-1-neg 64.6%

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

      8. unpow2 64.6%

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

      9. +-commutative 64.6%

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

      10. unpow2 64.6%

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

      11. fma-udef 64.6%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in l around inf 55.5%

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

   if 2.0000000000000001e-196 < t < 1.34999999999999991e-155

   1. Initial program 12.8%

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

      1. associate-*r/ 12.8%

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

      2. fma-neg 12.8%

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

      3. remove-double-neg 12.8%

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

      4. fma-neg 12.8%

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

      5. sub-neg 12.8%

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

      6. metadata-eval 12.8%

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

      7. remove-double-neg 12.8%

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

      8. +-commutative 12.8%

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

      9. fma-def 12.8%

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

   3. Simplified 12.8%

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

   5. Applied egg-rr 47.8%

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

   6. Taylor expanded in t around -inf 1.6%

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

      1. mul-1-neg 1.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.2%

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

   11. Simplified 79.2%

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

   if 1.34999999999999991e-155 < t < 0.0058

   1. Initial program 59.5%

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

      1. associate-*r/ 59.4%

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

      2. fma-neg 59.3%

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

      3. remove-double-neg 59.3%

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

      4. fma-neg 59.4%

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

      5. sub-neg 59.4%

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

      6. metadata-eval 59.4%

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

      7. remove-double-neg 59.4%

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

      8. +-commutative 59.4%

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

      9. fma-def 59.4%

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

   3. Simplified 59.4%

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

   4. Taylor expanded in x around inf 81.3%

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

      1. associate--l+ 81.5%

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

      2. unpow2 81.5%

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

      3. distribute-lft-out 81.5%

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

      4. unpow2 81.5%

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

      5. unpow2 81.5%

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

      6. associate-*r/ 81.5%

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

      7. mul-1-neg 81.5%

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

      8. unpow2 81.5%

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

      9. +-commutative 81.5%

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

      10. unpow2 81.5%

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

      11. fma-udef 81.5%

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

   6. Simplified 81.3%

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

   if 0.0058 < t

   1. Initial program 39.5%

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

      1. associate-*r/ 39.4%

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

      2. fma-neg 39.4%

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

      3. remove-double-neg 39.4%

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

      4. fma-neg 39.4%

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

      5. sub-neg 39.4%

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

      6. metadata-eval 39.4%

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

      7. remove-double-neg 39.4%

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

      8. +-commutative 39.4%

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

      9. fma-def 39.4%

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

   3. Simplified 39.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in t around inf 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-33}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0058:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0054:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt
            (+
             (/ (* l l) x)
             (+
              (* 2.0 (+ (* t t) (/ (* t t) x)))
              (/ (fma 2.0 (* t t) (* l l)) x)))))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1.45e-31)
     (- t_2)
     (if (<= t -4.3e-253)
       t_1
       (if (<= t -4e-305)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 1.7e-191)
           (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x)))))
           (if (<= t 1.35e-155) 1.0 (if (<= t 0.0054) t_1 t_2))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / sqrt((((l * l) / x) + ((2.0 * ((t * t) + ((t * t) / x))) + (fma(2.0, (t * t), (l * l)) / x)))));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1.45e-31) {
		tmp = -t_2;
	} else if (t <= -4.3e-253) {
		tmp = t_1;
	} else if (t <= -4e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.7e-191) {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.0054) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(Float64(Float64(l * l) / x) + Float64(Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))) + Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x))))))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1.45e-31)
		tmp = Float64(-t_2);
	elseif (t <= -4.3e-253)
		tmp = t_1;
	elseif (t <= -4e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.7e-191)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.0054)
		tmp = t_1;
	else
		tmp = 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[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision] + N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.45e-31], (-t$95$2), If[LessEqual[t, -4.3e-253], t$95$1, If[LessEqual[t, -4e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.7e-191], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-155], 1.0, If[LessEqual[t, 0.0054], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.45e-31

   1. Initial program 42.5%

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

      1. associate-*r/ 42.4%

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

      2. fma-neg 42.4%

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

      3. remove-double-neg 42.4%

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

      4. fma-neg 42.4%

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

      5. sub-neg 42.4%

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

      6. metadata-eval 42.4%

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

      7. remove-double-neg 42.4%

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

      8. +-commutative 42.4%

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

      9. fma-def 42.4%

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

   3. Simplified 42.4%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 92.5%

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

      3. metadata-eval 92.5%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 92.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.45e-31 < t < -4.3000000000000002e-253 or 1.34999999999999991e-155 < t < 0.0054000000000000003

   1. Initial program 43.5%

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

      1. associate-*l/ 43.5%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in x around inf 77.5%

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

      1. associate--l+ 77.5%

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

      2. unpow2 77.5%

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

      3. distribute-lft-out 77.5%

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

      4. unpow2 77.5%

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

      5. unpow2 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unpow2 77.5%

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

      9. +-commutative 77.5%

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

      10. unpow2 77.5%

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

      11. fma-udef 77.5%

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

   6. Simplified 77.5%

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

   if -4.3000000000000002e-253 < t < -3.99999999999999999e-305

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. remove-double-neg 2.9%

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

      4. fma-neg 2.9%

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

      5. sub-neg 2.9%

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

      6. metadata-eval 2.9%

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

      7. remove-double-neg 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-def 2.9%

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

   3. Simplified 2.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in t around -inf 67.0%

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

      1. mul-1-neg 67.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 67.0%

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

      3. metadata-eval 67.0%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 67.0%

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

   if -3.99999999999999999e-305 < t < 1.69999999999999997e-191

   1. Initial program 4.3%

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

      1. associate-*r/ 4.3%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.3%

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

      5. sub-neg 4.3%

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

      6. metadata-eval 4.3%

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

      7. remove-double-neg 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-def 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in x around inf 64.7%

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

      1. associate--l+ 64.6%

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

      2. unpow2 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. unpow2 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r/ 64.6%

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

      7. mul-1-neg 64.6%

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

      8. unpow2 64.6%

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

      9. +-commutative 64.6%

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

      10. unpow2 64.6%

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

      11. fma-udef 64.6%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in l around inf 55.5%

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

   if 1.69999999999999997e-191 < t < 1.34999999999999991e-155

   1. Initial program 12.8%

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

      1. associate-*r/ 12.8%

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

      2. fma-neg 12.8%

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

      3. remove-double-neg 12.8%

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

      4. fma-neg 12.8%

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

      5. sub-neg 12.8%

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

      6. metadata-eval 12.8%

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

      7. remove-double-neg 12.8%

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

      8. +-commutative 12.8%

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

      9. fma-def 12.8%

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

   3. Simplified 12.8%

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

   5. Applied egg-rr 47.8%

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

   6. Taylor expanded in t around -inf 1.6%

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

      1. mul-1-neg 1.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.2%

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

   11. Simplified 79.2%

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

   if 0.0054000000000000003 < t

   1. Initial program 39.5%

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

      1. associate-*r/ 39.4%

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

      2. fma-neg 39.4%

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

      3. remove-double-neg 39.4%

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

      4. fma-neg 39.4%

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

      5. sub-neg 39.4%

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

      6. metadata-eval 39.4%

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

      7. remove-double-neg 39.4%

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

      8. +-commutative 39.4%

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

      9. fma-def 39.4%

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

   3. Simplified 39.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in t around inf 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0054:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{2} \cdot \frac{t}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-31}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-259}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0045:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_2 (/ (* l l) x))
        (t_3
         (*
          (sqrt 2.0)
          (/ t (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))))))))
   (if (<= t -1.4e-31)
     (- t_1)
     (if (<= t -4.7e-259)
       t_3
       (if (<= t -4e-305)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 9.8e-191)
           (* (sqrt 2.0) (/ t (* (sqrt 2.0) (* l (sqrt (/ 1.0 x))))))
           (if (<= t 1.35e-155) 1.0 (if (<= t 0.0045) t_3 t_1))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = sqrt(2.0) * (t / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.4e-31) {
		tmp = -t_1;
	} else if (t <= -4.7e-259) {
		tmp = t_3;
	} else if (t <= -4e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 9.8e-191) {
		tmp = sqrt(2.0) * (t / (sqrt(2.0) * (l * sqrt((1.0 / x)))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.0045) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_2 = (l * l) / x
    t_3 = sqrt(2.0d0) * (t / sqrt((t_2 + (t_2 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    if (t <= (-1.4d-31)) then
        tmp = -t_1
    else if (t <= (-4.7d-259)) then
        tmp = t_3
    else if (t <= (-4d-305)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 9.8d-191) then
        tmp = sqrt(2.0d0) * (t / (sqrt(2.0d0) * (l * sqrt((1.0d0 / x)))))
    else if (t <= 1.35d-155) then
        tmp = 1.0d0
    else if (t <= 0.0045d0) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = Math.sqrt(2.0) * (t / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.4e-31) {
		tmp = -t_1;
	} else if (t <= -4.7e-259) {
		tmp = t_3;
	} else if (t <= -4e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 9.8e-191) {
		tmp = Math.sqrt(2.0) * (t / (Math.sqrt(2.0) * (l * Math.sqrt((1.0 / x)))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.0045) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_2 = (l * l) / x
	t_3 = math.sqrt(2.0) * (t / math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))))
	tmp = 0
	if t <= -1.4e-31:
		tmp = -t_1
	elif t <= -4.7e-259:
		tmp = t_3
	elif t <= -4e-305:
		tmp = (1.0 / x) + -1.0
	elif t <= 9.8e-191:
		tmp = math.sqrt(2.0) * (t / (math.sqrt(2.0) * (l * math.sqrt((1.0 / x)))))
	elif t <= 1.35e-155:
		tmp = 1.0
	elif t <= 0.0045:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))))
	tmp = 0.0
	if (t <= -1.4e-31)
		tmp = Float64(-t_1);
	elseif (t <= -4.7e-259)
		tmp = t_3;
	elseif (t <= -4e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 9.8e-191)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(sqrt(2.0) * Float64(l * sqrt(Float64(1.0 / x))))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.0045)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	t_2 = (l * l) / x;
	t_3 = sqrt(2.0) * (t / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	tmp = 0.0;
	if (t <= -1.4e-31)
		tmp = -t_1;
	elseif (t <= -4.7e-259)
		tmp = t_3;
	elseif (t <= -4e-305)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 9.8e-191)
		tmp = sqrt(2.0) * (t / (sqrt(2.0) * (l * sqrt((1.0 / x)))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.0045)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.4e-31], (-t$95$1), If[LessEqual[t, -4.7e-259], t$95$3, If[LessEqual[t, -4e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 9.8e-191], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-155], 1.0, If[LessEqual[t, 0.0045], t$95$3, t$95$1]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.3999999999999999e-31

   1. Initial program 42.5%

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

      1. associate-*r/ 42.4%

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

      2. fma-neg 42.4%

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

      3. remove-double-neg 42.4%

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

      4. fma-neg 42.4%

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

      5. sub-neg 42.4%

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

      6. metadata-eval 42.4%

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

      7. remove-double-neg 42.4%

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

      8. +-commutative 42.4%

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

      9. fma-def 42.4%

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

   3. Simplified 42.4%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 92.5%

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

      3. metadata-eval 92.5%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 92.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.3999999999999999e-31 < t < -4.69999999999999998e-259 or 1.34999999999999991e-155 < t < 0.00449999999999999966

   1. Initial program 43.5%

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

      1. associate-*r/ 43.4%

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

      2. fma-neg 43.4%

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

      3. remove-double-neg 43.4%

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

      4. fma-neg 43.4%

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

      5. sub-neg 43.4%

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

      6. metadata-eval 43.4%

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

      7. remove-double-neg 43.4%

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

      8. +-commutative 43.4%

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

      9. fma-def 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in x around inf 77.4%

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

      1. associate--l+ 77.5%

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

      2. unpow2 77.5%

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

      3. distribute-lft-out 77.5%

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

      4. unpow2 77.5%

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

      5. unpow2 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unpow2 77.5%

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

      9. +-commutative 77.5%

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

      10. unpow2 77.5%

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

      11. fma-udef 77.5%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unpow2 77.3%

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

      4. distribute-rgt-neg-in 77.3%

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

   9. Simplified 77.3%

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

   if -4.69999999999999998e-259 < t < -3.99999999999999999e-305

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. remove-double-neg 2.9%

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

      4. fma-neg 2.9%

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

      5. sub-neg 2.9%

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

      6. metadata-eval 2.9%

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

      7. remove-double-neg 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-def 2.9%

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

   3. Simplified 2.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in t around -inf 67.0%

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

      1. mul-1-neg 67.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 67.0%

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

      3. metadata-eval 67.0%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 67.0%

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

   if -3.99999999999999999e-305 < t < 9.7999999999999999e-191

   1. Initial program 4.3%

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

      1. associate-*r/ 4.3%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.3%

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

      5. sub-neg 4.3%

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

      6. metadata-eval 4.3%

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

      7. remove-double-neg 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-def 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in x around inf 64.7%

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

      1. associate--l+ 64.6%

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

      2. unpow2 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. unpow2 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r/ 64.6%

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

      7. mul-1-neg 64.6%

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

      8. unpow2 64.6%

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

      9. +-commutative 64.6%

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

      10. unpow2 64.6%

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

      11. fma-udef 64.6%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in l around inf 55.5%

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

      1. associate-*l* 55.5%

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

   9. Simplified 55.5%

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

   if 9.7999999999999999e-191 < t < 1.34999999999999991e-155

   1. Initial program 12.8%

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

      1. associate-*r/ 12.8%

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

      2. fma-neg 12.8%

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

      3. remove-double-neg 12.8%

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

      4. fma-neg 12.8%

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

      5. sub-neg 12.8%

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

      6. metadata-eval 12.8%

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

      7. remove-double-neg 12.8%

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

      8. +-commutative 12.8%

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

      9. fma-def 12.8%

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

   3. Simplified 12.8%

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

   5. Applied egg-rr 47.8%

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

   6. Taylor expanded in t around -inf 1.6%

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

      1. mul-1-neg 1.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.2%

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

   11. Simplified 79.2%

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

   if 0.00449999999999999966 < t

   1. Initial program 39.5%

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

      1. associate-*r/ 39.4%

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

      2. fma-neg 39.4%

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

      3. remove-double-neg 39.4%

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

      4. fma-neg 39.4%

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

      5. sub-neg 39.4%

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

      6. metadata-eval 39.4%

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

      7. remove-double-neg 39.4%

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

      8. +-commutative 39.4%

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

      9. fma-def 39.4%

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

   3. Simplified 39.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in t around inf 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-31}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-259}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-191}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0045:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 8: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{2} \cdot \frac{t}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-262}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0019:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_2 (/ (* l l) x))
        (t_3
         (*
          (sqrt 2.0)
          (/ t (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))))))))
   (if (<= t -1.45e-31)
     (- t_1)
     (if (<= t -7.8e-262)
       t_3
       (if (<= t -1.45e-305)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 1.05e-199)
           (* (sqrt 2.0) (/ t (* (* (sqrt 2.0) l) (sqrt (/ 1.0 x)))))
           (if (<= t 1.35e-155) 1.0 (if (<= t 0.0019) t_3 t_1))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = sqrt(2.0) * (t / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.45e-31) {
		tmp = -t_1;
	} else if (t <= -7.8e-262) {
		tmp = t_3;
	} else if (t <= -1.45e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.05e-199) {
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.0019) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_2 = (l * l) / x
    t_3 = sqrt(2.0d0) * (t / sqrt((t_2 + (t_2 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    if (t <= (-1.45d-31)) then
        tmp = -t_1
    else if (t <= (-7.8d-262)) then
        tmp = t_3
    else if (t <= (-1.45d-305)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 1.05d-199) then
        tmp = sqrt(2.0d0) * (t / ((sqrt(2.0d0) * l) * sqrt((1.0d0 / x))))
    else if (t <= 1.35d-155) then
        tmp = 1.0d0
    else if (t <= 0.0019d0) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = Math.sqrt(2.0) * (t / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.45e-31) {
		tmp = -t_1;
	} else if (t <= -7.8e-262) {
		tmp = t_3;
	} else if (t <= -1.45e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.05e-199) {
		tmp = Math.sqrt(2.0) * (t / ((Math.sqrt(2.0) * l) * Math.sqrt((1.0 / x))));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.0019) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_2 = (l * l) / x
	t_3 = math.sqrt(2.0) * (t / math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))))
	tmp = 0
	if t <= -1.45e-31:
		tmp = -t_1
	elif t <= -7.8e-262:
		tmp = t_3
	elif t <= -1.45e-305:
		tmp = (1.0 / x) + -1.0
	elif t <= 1.05e-199:
		tmp = math.sqrt(2.0) * (t / ((math.sqrt(2.0) * l) * math.sqrt((1.0 / x))))
	elif t <= 1.35e-155:
		tmp = 1.0
	elif t <= 0.0019:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))))
	tmp = 0.0
	if (t <= -1.45e-31)
		tmp = Float64(-t_1);
	elseif (t <= -7.8e-262)
		tmp = t_3;
	elseif (t <= -1.45e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.05e-199)
		tmp = Float64(sqrt(2.0) * Float64(t / Float64(Float64(sqrt(2.0) * l) * sqrt(Float64(1.0 / x)))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.0019)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	t_2 = (l * l) / x;
	t_3 = sqrt(2.0) * (t / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	tmp = 0.0;
	if (t <= -1.45e-31)
		tmp = -t_1;
	elseif (t <= -7.8e-262)
		tmp = t_3;
	elseif (t <= -1.45e-305)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 1.05e-199)
		tmp = sqrt(2.0) * (t / ((sqrt(2.0) * l) * sqrt((1.0 / x))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.0019)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.45e-31], (-t$95$1), If[LessEqual[t, -7.8e-262], t$95$3, If[LessEqual[t, -1.45e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.05e-199], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[(N[(N[Sqrt[2.0], $MachinePrecision] * l), $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-155], 1.0, If[LessEqual[t, 0.0019], t$95$3, t$95$1]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.45e-31

   1. Initial program 42.5%

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

      1. associate-*r/ 42.4%

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

      2. fma-neg 42.4%

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

      3. remove-double-neg 42.4%

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

      4. fma-neg 42.4%

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

      5. sub-neg 42.4%

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

      6. metadata-eval 42.4%

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

      7. remove-double-neg 42.4%

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

      8. +-commutative 42.4%

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

      9. fma-def 42.4%

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

   3. Simplified 42.4%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 92.5%

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

      3. metadata-eval 92.5%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 92.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.45e-31 < t < -7.79999999999999967e-262 or 1.34999999999999991e-155 < t < 0.0019

   1. Initial program 43.5%

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

      1. associate-*r/ 43.4%

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

      2. fma-neg 43.4%

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

      3. remove-double-neg 43.4%

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

      4. fma-neg 43.4%

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

      5. sub-neg 43.4%

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

      6. metadata-eval 43.4%

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

      7. remove-double-neg 43.4%

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

      8. +-commutative 43.4%

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

      9. fma-def 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in x around inf 77.4%

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

      1. associate--l+ 77.5%

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

      2. unpow2 77.5%

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

      3. distribute-lft-out 77.5%

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

      4. unpow2 77.5%

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

      5. unpow2 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unpow2 77.5%

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

      9. +-commutative 77.5%

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

      10. unpow2 77.5%

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

      11. fma-udef 77.5%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unpow2 77.3%

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

      4. distribute-rgt-neg-in 77.3%

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

   9. Simplified 77.3%

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

   if -7.79999999999999967e-262 < t < -1.44999999999999994e-305

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. remove-double-neg 2.9%

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

      4. fma-neg 2.9%

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

      5. sub-neg 2.9%

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

      6. metadata-eval 2.9%

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

      7. remove-double-neg 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-def 2.9%

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

   3. Simplified 2.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in t around -inf 67.0%

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

      1. mul-1-neg 67.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 67.0%

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

      3. metadata-eval 67.0%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 67.0%

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

   if -1.44999999999999994e-305 < t < 1.05000000000000001e-199

   1. Initial program 4.3%

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

      1. associate-*r/ 4.3%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.3%

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

      5. sub-neg 4.3%

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

      6. metadata-eval 4.3%

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

      7. remove-double-neg 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-def 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in x around inf 64.7%

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

      1. associate--l+ 64.6%

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

      2. unpow2 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. unpow2 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r/ 64.6%

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

      7. mul-1-neg 64.6%

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

      8. unpow2 64.6%

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

      9. +-commutative 64.6%

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

      10. unpow2 64.6%

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

      11. fma-udef 64.6%

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

   6. Simplified 64.7%

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

   7. Taylor expanded in l around inf 55.5%

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

   if 1.05000000000000001e-199 < t < 1.34999999999999991e-155

   1. Initial program 12.8%

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

      1. associate-*r/ 12.8%

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

      2. fma-neg 12.8%

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

      3. remove-double-neg 12.8%

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

      4. fma-neg 12.8%

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

      5. sub-neg 12.8%

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

      6. metadata-eval 12.8%

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

      7. remove-double-neg 12.8%

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

      8. +-commutative 12.8%

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

      9. fma-def 12.8%

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

   3. Simplified 12.8%

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

   5. Applied egg-rr 47.8%

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

   6. Taylor expanded in t around -inf 1.6%

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

      1. mul-1-neg 1.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

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

       2. rem-square-sqrt 79.2%

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

   11. Simplified 79.2%

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

   if 0.0019 < t

   1. Initial program 39.5%

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

      1. associate-*r/ 39.4%

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

      2. fma-neg 39.4%

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

      3. remove-double-neg 39.4%

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

      4. fma-neg 39.4%

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

      5. sub-neg 39.4%

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

      6. metadata-eval 39.4%

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

      7. remove-double-neg 39.4%

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

      8. +-commutative 39.4%

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

      9. fma-def 39.4%

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

   3. Simplified 39.4%

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

   5. Applied egg-rr 84.4%

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

   6. Taylor expanded in t around inf 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-262}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-199}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.0019:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 9: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{2} \cdot \frac{t}{\sqrt{t_2 + \left(t_2 + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-255}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{t_2 + t_2}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.00192:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_2 (/ (* l l) x))
        (t_3
         (*
          (sqrt 2.0)
          (/ t (sqrt (+ t_2 (+ t_2 (* 2.0 (+ (* t t) (/ (* t t) x))))))))))
   (if (<= t -1.45e-31)
     (- t_1)
     (if (<= t -1.65e-255)
       t_3
       (if (<= t -1.45e-305)
         (+ (/ 1.0 x) -1.0)
         (if (<= t 1.12e-198)
           (* (sqrt 2.0) (/ t (sqrt (+ t_2 t_2))))
           (if (<= t 1.35e-155) 1.0 (if (<= t 0.00192) t_3 t_1))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = sqrt(2.0) * (t / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.45e-31) {
		tmp = -t_1;
	} else if (t <= -1.65e-255) {
		tmp = t_3;
	} else if (t <= -1.45e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.12e-198) {
		tmp = sqrt(2.0) * (t / sqrt((t_2 + t_2)));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.00192) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_2 = (l * l) / x
    t_3 = sqrt(2.0d0) * (t / sqrt((t_2 + (t_2 + (2.0d0 * ((t * t) + ((t * t) / x)))))))
    if (t <= (-1.45d-31)) then
        tmp = -t_1
    else if (t <= (-1.65d-255)) then
        tmp = t_3
    else if (t <= (-1.45d-305)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 1.12d-198) then
        tmp = sqrt(2.0d0) * (t / sqrt((t_2 + t_2)))
    else if (t <= 1.35d-155) then
        tmp = 1.0d0
    else if (t <= 0.00192d0) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = Math.sqrt(2.0) * (t / Math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	double tmp;
	if (t <= -1.45e-31) {
		tmp = -t_1;
	} else if (t <= -1.65e-255) {
		tmp = t_3;
	} else if (t <= -1.45e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 1.12e-198) {
		tmp = Math.sqrt(2.0) * (t / Math.sqrt((t_2 + t_2)));
	} else if (t <= 1.35e-155) {
		tmp = 1.0;
	} else if (t <= 0.00192) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_2 = (l * l) / x
	t_3 = math.sqrt(2.0) * (t / math.sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))))
	tmp = 0
	if t <= -1.45e-31:
		tmp = -t_1
	elif t <= -1.65e-255:
		tmp = t_3
	elif t <= -1.45e-305:
		tmp = (1.0 / x) + -1.0
	elif t <= 1.12e-198:
		tmp = math.sqrt(2.0) * (t / math.sqrt((t_2 + t_2)))
	elif t <= 1.35e-155:
		tmp = 1.0
	elif t <= 0.00192:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_2 + Float64(t_2 + Float64(2.0 * Float64(Float64(t * t) + Float64(Float64(t * t) / x))))))))
	tmp = 0.0
	if (t <= -1.45e-31)
		tmp = Float64(-t_1);
	elseif (t <= -1.65e-255)
		tmp = t_3;
	elseif (t <= -1.45e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 1.12e-198)
		tmp = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_2 + t_2))));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.00192)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	t_2 = (l * l) / x;
	t_3 = sqrt(2.0) * (t / sqrt((t_2 + (t_2 + (2.0 * ((t * t) + ((t * t) / x)))))));
	tmp = 0.0;
	if (t <= -1.45e-31)
		tmp = -t_1;
	elseif (t <= -1.65e-255)
		tmp = t_3;
	elseif (t <= -1.45e-305)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 1.12e-198)
		tmp = sqrt(2.0) * (t / sqrt((t_2 + t_2)));
	elseif (t <= 1.35e-155)
		tmp = 1.0;
	elseif (t <= 0.00192)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$2 + N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.45e-31], (-t$95$1), If[LessEqual[t, -1.65e-255], t$95$3, If[LessEqual[t, -1.45e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 1.12e-198], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$2 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.35e-155], 1.0, If[LessEqual[t, 0.00192], t$95$3, t$95$1]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.45e-31

   1. Initial program 42.5%

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

      1. associate-*r/ 42.4%

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

      2. fma-neg 42.4%

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

      3. remove-double-neg 42.4%

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

      4. fma-neg 42.4%

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

      5. sub-neg 42.4%

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

      6. metadata-eval 42.4%

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

      7. remove-double-neg 42.4%

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

      8. +-commutative 42.4%

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

      9. fma-def 42.4%

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

   3. Simplified 42.4%

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

   5. Applied egg-rr 84.5%

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

   6. Taylor expanded in t around -inf 92.5%

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

      1. mul-1-neg 92.5%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 92.5%

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

      3. metadata-eval 92.5%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 92.5%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.45e-31 < t < -1.64999999999999994e-255 or 1.34999999999999991e-155 < t < 0.00192000000000000005

   1. Initial program 43.5%

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

      1. associate-*r/ 43.4%

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

      2. fma-neg 43.4%

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

      3. remove-double-neg 43.4%

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

      4. fma-neg 43.4%

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

      5. sub-neg 43.4%

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

      6. metadata-eval 43.4%

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

      7. remove-double-neg 43.4%

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

      8. +-commutative 43.4%

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

      9. fma-def 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in x around inf 77.4%

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

      1. associate--l+ 77.5%

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

      2. unpow2 77.5%

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

      3. distribute-lft-out 77.5%

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

      4. unpow2 77.5%

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

      5. unpow2 77.5%

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

      6. associate-*r/ 77.5%

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

      7. mul-1-neg 77.5%

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

      8. unpow2 77.5%

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

      9. +-commutative 77.5%

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

      10. unpow2 77.5%

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

      11. fma-udef 77.5%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in t around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unpow2 77.3%

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

      4. distribute-rgt-neg-in 77.3%

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

   9. Simplified 77.3%

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

   if -1.64999999999999994e-255 < t < -1.44999999999999994e-305

   1. Initial program 2.9%

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

      1. associate-*r/ 2.9%

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

      2. fma-neg 2.9%

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

      3. remove-double-neg 2.9%

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

      4. fma-neg 2.9%

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

      5. sub-neg 2.9%

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

      6. metadata-eval 2.9%

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

      7. remove-double-neg 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-def 2.9%

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

   3. Simplified 2.9%

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

   5. Applied egg-rr 30.6%

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

   6. Taylor expanded in t around -inf 67.0%

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

      1. mul-1-neg 67.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 67.0%

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

      3. metadata-eval 67.0%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 67.0%

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

   if -1.44999999999999994e-305 < t < 1.12000000000000007e-198

   1. Initial program 4.3%

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

      1. associate-*r/ 4.3%

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

      2. fma-neg 4.2%

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

      3. remove-double-neg 4.2%

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

      4. fma-neg 4.3%

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

      5. sub-neg 4.3%

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

      6. metadata-eval 4.3%

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

      7. remove-double-neg 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-def 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in x around inf 64.7%

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

      1. associate--l+ 64.6%

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

      2. unpow2 64.6%

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

      3. distribute-lft-out 64.6%

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

      4. unpow2 64.6%

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

      5. unpow2 64.6%

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

      6. associate-*r/ 64.6%

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

      7. mul-1-neg 64.6%

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

      8. unpow2 64.6%

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

      9. +-commutative 64.6%

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

      10. unpow2 64.6%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef 64.6%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

   6. Simplified 64.7%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]

   7. Taylor expanded in t around 0 64.7%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \]
   8. Step-by-step derivation

      1. sub-neg 64.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(--1 \cdot \frac{{\ell}^{2}}{x}\right)}}} \]

      2. unpow2 64.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(--1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]

      3. mul-1-neg 64.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(-\color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]

      4. remove-double-neg 64.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \color{blue}{\frac{{\ell}^{2}}{x}}}} \]

      5. unpow2 64.7%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\color{blue}{\ell \cdot \ell}}{x}}} \]

   9. Simplified 64.7%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}} \]

   if 1.12000000000000007e-198 < t < 1.34999999999999991e-155

   1. Initial program 12.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 12.8%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 12.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 12.8%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 47.8%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 1.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 1.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto -\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

       2. rem-square-sqrt 79.2%

          \[\leadsto -\color{blue}{-1} \]

   11. Simplified 79.2%

       \[\leadsto -\color{blue}{-1} \]

   if 0.00192000000000000005 < t

   1. Initial program 39.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 39.4%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 39.4%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 39.4%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around inf 95.1%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-255}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-155}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 0.00192:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\frac{\ell \cdot \ell}{x} + 2 \cdot \left(t \cdot t + \frac{t \cdot t}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 10: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ t_3 := \sqrt{2} \cdot \frac{t}{\sqrt{t_2 + t_2}}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-200}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_2 (/ (* l l) x))
        (t_3 (* (sqrt 2.0) (/ t (sqrt (+ t_2 t_2))))))
   (if (<= t -4.5e-130)
     (- t_1)
     (if (<= t -2.85e-254)
       t_3
       (if (<= t -4e-305) (+ (/ 1.0 x) -1.0) (if (<= t 7.4e-200) t_3 t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = sqrt(2.0) * (t / sqrt((t_2 + t_2)));
	double tmp;
	if (t <= -4.5e-130) {
		tmp = -t_1;
	} else if (t <= -2.85e-254) {
		tmp = t_3;
	} else if (t <= -4e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 7.4e-200) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    t_2 = (l * l) / x
    t_3 = sqrt(2.0d0) * (t / sqrt((t_2 + t_2)))
    if (t <= (-4.5d-130)) then
        tmp = -t_1
    else if (t <= (-2.85d-254)) then
        tmp = t_3
    else if (t <= (-4d-305)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else if (t <= 7.4d-200) then
        tmp = t_3
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double t_3 = Math.sqrt(2.0) * (t / Math.sqrt((t_2 + t_2)));
	double tmp;
	if (t <= -4.5e-130) {
		tmp = -t_1;
	} else if (t <= -2.85e-254) {
		tmp = t_3;
	} else if (t <= -4e-305) {
		tmp = (1.0 / x) + -1.0;
	} else if (t <= 7.4e-200) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	t_2 = (l * l) / x
	t_3 = math.sqrt(2.0) * (t / math.sqrt((t_2 + t_2)))
	tmp = 0
	if t <= -4.5e-130:
		tmp = -t_1
	elif t <= -2.85e-254:
		tmp = t_3
	elif t <= -4e-305:
		tmp = (1.0 / x) + -1.0
	elif t <= 7.4e-200:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(Float64(l * l) / x)
	t_3 = Float64(sqrt(2.0) * Float64(t / sqrt(Float64(t_2 + t_2))))
	tmp = 0.0
	if (t <= -4.5e-130)
		tmp = Float64(-t_1);
	elseif (t <= -2.85e-254)
		tmp = t_3;
	elseif (t <= -4e-305)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	elseif (t <= 7.4e-200)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	t_2 = (l * l) / x;
	t_3 = sqrt(2.0) * (t / sqrt((t_2 + t_2)));
	tmp = 0.0;
	if (t <= -4.5e-130)
		tmp = -t_1;
	elseif (t <= -2.85e-254)
		tmp = t_3;
	elseif (t <= -4e-305)
		tmp = (1.0 / x) + -1.0;
	elseif (t <= 7.4e-200)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[2.0], $MachinePrecision] * N[(t / N[Sqrt[N[(t$95$2 + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.5e-130], (-t$95$1), If[LessEqual[t, -2.85e-254], t$95$3, If[LessEqual[t, -4e-305], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t, 7.4e-200], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.5e-130

   1. Initial program 44.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 44.0%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 44.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 87.1%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 87.1%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 87.1%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 87.1%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 87.1%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -4.5e-130 < t < -2.8500000000000001e-254 or -3.99999999999999999e-305 < t < 7.40000000000000022e-200

   1. Initial program 9.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 9.5%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 9.5%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 9.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   4. Taylor expanded in x around inf 67.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \]
   5. Step-by-step derivation

      1. associate--l+ 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      2. unpow2 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(\left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      3. distribute-lft-out 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(\color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. unpow2 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{\color{blue}{t \cdot t}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + \color{blue}{t \cdot t}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. associate-*r/ 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

      7. mul-1-neg 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{\color{blue}{-\left({\ell}^{2} + 2 \cdot {t}^{2}\right)}}{x}\right)}} \cdot t \]

      8. unpow2 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(\color{blue}{\ell \cdot \ell} + 2 \cdot {t}^{2}\right)}{x}\right)}} \cdot t \]

      9. +-commutative 67.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\left(2 \cdot {t}^{2} + \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

      10. unpow2 67.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\left(2 \cdot \color{blue}{\left(t \cdot t\right)} + \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

      11. fma-udef 67.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}{x}\right)}} \cdot t \]

   6. Simplified 67.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\frac{t \cdot t}{x} + t \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \]

   7. Taylor expanded in t around 0 59.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}} \]
   8. Step-by-step derivation

      1. sub-neg 59.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(--1 \cdot \frac{{\ell}^{2}}{x}\right)}}} \]

      2. unpow2 59.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(--1 \cdot \frac{{\ell}^{2}}{x}\right)}} \]

      3. mul-1-neg 59.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \left(-\color{blue}{\left(-\frac{{\ell}^{2}}{x}\right)}\right)}} \]

      4. remove-double-neg 59.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \color{blue}{\frac{{\ell}^{2}}{x}}}} \]

      5. unpow2 59.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\color{blue}{\ell \cdot \ell}}{x}}} \]

   9. Simplified 59.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}} \]

   if -2.8500000000000001e-254 < t < -3.99999999999999999e-305

   1. Initial program 2.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 2.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 2.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 2.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 67.0%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 67.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 67.0%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 67.0%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 67.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 67.0%

      \[\leadsto -\color{blue}{\left(1 - \frac{1}{x}\right)} \]

   if 7.40000000000000022e-200 < t

   1. Initial program 43.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 43.2%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 43.2%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 43.2%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around inf 85.0%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -2.85 \cdot 10^{-254}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{\ell \cdot \ell}{x} + \frac{\ell \cdot \ell}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 11: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-t_1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -1e-309) (- t_1) t_1)))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1e-309) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-1d-309)) then
        tmp = -t_1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -1e-309) {
		tmp = -t_1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -1e-309:
		tmp = -t_1
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-t_1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -t_1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1e-309], (-t$95$1), t$95$1]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 35.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 74.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 74.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 74.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   if -1.000000000000002e-309 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around inf 77.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 12: 76.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309)
   (+ -1.0 (- (/ 1.0 x) (/ 0.5 (* x x))))
   (sqrt (/ (+ x -1.0) (+ x 1.0)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else {
		tmp = sqrt(((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 <= (-1d-309)) then
        tmp = (-1.0d0) + ((1.0d0 / x) - (0.5d0 / (x * x)))
    else
        tmp = sqrt(((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 <= -1e-309) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)))
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))));
	else
		tmp = sqrt(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 <= -1e-309)
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	else
		tmp = sqrt(((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, -1e-309], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 35.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 74.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 74.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 74.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 73.8%

      \[\leadsto -\color{blue}{\left(\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 73.8%

          \[\leadsto -\color{blue}{\left(1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)\right)} \]

       2. associate-*r/ 73.8%

          \[\leadsto -\left(1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right)\right) \]

       3. metadata-eval 73.8%

          \[\leadsto -\left(1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right)\right) \]

       4. unpow2 73.8%

          \[\leadsto -\left(1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right)\right) \]

   11. Simplified 73.8%

       \[\leadsto -\color{blue}{\left(1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\right)} \]

   if -1.000000000000002e-309 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around inf 77.4%

      \[\leadsto \color{blue}{\sqrt{\frac{x - 1}{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 13: 75.7% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (+ -1.0 (- (/ 1.0 x) (/ 0.5 (* x x)))) (- 1.0 (/ 1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	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 <= (-1d-309)) then
        tmp = (-1.0d0) + ((1.0d0 / x) - (0.5d0 / (x * x)))
    else
        tmp = 1.0d0 - (1.0d0 / x)
    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 <= -1e-309) {
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)))
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(-1.0 + Float64(Float64(1.0 / x) - Float64(0.5 / Float64(x * x))));
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0 + ((1.0 / x) - (0.5 / (x * x)));
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(-1.0 + N[(N[(1.0 / x), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 35.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 74.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 74.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 74.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 73.8%

      \[\leadsto -\color{blue}{\left(\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. associate--l+ 73.8%

          \[\leadsto -\color{blue}{\left(1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)\right)} \]

       2. associate-*r/ 73.8%

          \[\leadsto -\left(1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right)\right) \]

       3. metadata-eval 73.8%

          \[\leadsto -\left(1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right)\right) \]

       4. unpow2 73.8%

          \[\leadsto -\left(1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right)\right) \]

   11. Simplified 73.8%

       \[\leadsto -\color{blue}{\left(1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)\right)} \]

   if -1.000000000000002e-309 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.7%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 1.7%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto -\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right) \]

       2. rem-square-sqrt 76.6%

          \[\leadsto -\left(\color{blue}{-1} + \frac{1}{x}\right) \]

   11. Simplified 76.6%

       \[\leadsto -\color{blue}{\left(-1 + \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1 + \left(\frac{1}{x} - \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 14: 75.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -1e-309) -1.0 (- 1.0 (/ 1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	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 <= (-1d-309)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 - (1.0d0 / x)
    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 <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], -1.0, N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 35.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 74.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 74.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 74.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 72.2%

      \[\leadsto -\color{blue}{1} \]

   if -1.000000000000002e-309 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.7%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 1.7%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto -\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right) \]

       2. rem-square-sqrt 76.6%

          \[\leadsto -\left(\color{blue}{-1} + \frac{1}{x}\right) \]

   11. Simplified 76.6%

       \[\leadsto -\color{blue}{\left(-1 + \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 15: 75.6% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (+ (/ 1.0 x) -1.0) (- 1.0 (/ 1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	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 <= (-1d-309)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else
        tmp = 1.0d0 - (1.0d0 / x)
    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 <= -1e-309) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = 1.0 - (1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = (1.0 / x) + -1.0
	else:
		tmp = 1.0 - (1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	else
		tmp = Float64(1.0 - Float64(1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = (1.0 / x) + -1.0;
	else
		tmp = 1.0 - (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 - N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 35.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 74.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 74.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 74.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 73.4%

      \[\leadsto -\color{blue}{\left(1 - \frac{1}{x}\right)} \]

   if -1.000000000000002e-309 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.7%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 1.7%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{\left({\left(\sqrt{-1}\right)}^{2} + \frac{1}{x}\right)} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto -\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} + \frac{1}{x}\right) \]

       2. rem-square-sqrt 76.6%

          \[\leadsto -\left(\color{blue}{-1} + \frac{1}{x}\right) \]

   11. Simplified 76.6%

       \[\leadsto -\color{blue}{\left(-1 + \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{1}{x}\\ \end{array} \]

Alternative 16: 74.9% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -1e-309) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 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 <= (-1d-309)) then
        tmp = -1.0d0
    else
        tmp = 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 <= -1e-309) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 35.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 34.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 34.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 74.6%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 74.6%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 74.6%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 74.6%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 74.6%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around inf 72.2%

      \[\leadsto -\color{blue}{1} \]

   if -1.000000000000002e-309 < t

   1. Initial program 39.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*r/ 38.9%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

      2. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

      3. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

      4. fma-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

      5. sub-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      6. metadata-eval 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

      7. remove-double-neg 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

      8. +-commutative 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

      9. fma-def 38.9%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   3. Simplified 38.9%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

   5. Applied egg-rr 71.2%

      \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

   6. Taylor expanded in t around -inf 1.7%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 1.7%

         \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

      2. sub-neg 1.7%

         \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

      3. metadata-eval 1.7%

         \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

   8. Simplified 1.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   9. Taylor expanded in x around -inf 0.0%

      \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 0.0%

          \[\leadsto -\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

       2. rem-square-sqrt 76.4%

          \[\leadsto -\color{blue}{-1} \]

   11. Simplified 76.4%

       \[\leadsto -\color{blue}{-1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17: 38.6% 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 37.1%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*r/ 37.0%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}} \]

   2. fma-neg 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}} \]

   3. remove-double-neg 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}} \]

   4. fma-neg 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}} \]

   5. sub-neg 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

   6. metadata-eval 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}} \]

   7. remove-double-neg 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}} \]

   8. +-commutative 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(2 \cdot \left(t \cdot t\right) + \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

   9. fma-def 37.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)} - \ell \cdot \ell}} \]

3. Simplified 37.0%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]

5. Applied egg-rr 67.3%

   \[\leadsto \color{blue}{\frac{t \cdot \sqrt{2}}{\mathsf{hypot}\left(\sqrt{\frac{x + 1}{x + -1}} \cdot \mathsf{hypot}\left(\ell, t \cdot \sqrt{2}\right), \ell\right)}} \]

6. Taylor expanded in t around -inf 36.2%

   \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{x - 1}{1 + x}}} \]
7. Step-by-step derivation

   1. mul-1-neg 36.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{x - 1}{1 + x}}} \]

   2. sub-neg 36.2%

      \[\leadsto -\sqrt{\frac{\color{blue}{x + \left(-1\right)}}{1 + x}} \]

   3. metadata-eval 36.2%

      \[\leadsto -\sqrt{\frac{x + \color{blue}{-1}}{1 + x}} \]

8. Simplified 36.2%

   \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

9. Taylor expanded in x around -inf 0.0%

   \[\leadsto -\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \]
10. Step-by-step derivation

    1. unpow2 0.0%

       \[\leadsto -\color{blue}{\sqrt{-1} \cdot \sqrt{-1}} \]

    2. rem-square-sqrt 41.1%

       \[\leadsto -\color{blue}{-1} \]

11. Simplified 41.1%

    \[\leadsto -\color{blue}{-1} \]

12. Final simplification 41.1%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023216 
(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)))))