Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 87.2%

Time: 23.7s

Alternatives: 15

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: 32.9% 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: 87.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ t_3 := \frac{1}{x} + -1\\ \mathbf{if}\;t \leq -1 \cdot 10^{+146}:\\ \;\;\;\;t_3 - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-184}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-223}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{t_2 + t_2}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (fma (/ l x) l (+ (* 2.0 (* t (+ t (/ t x)))) (* l (/ l x))))))))
        (t_2 (fma 2.0 (* t t) (* l l)))
        (t_3 (+ (/ 1.0 x) -1.0)))
   (if (<= t -1e+146)
     (- t_3 (/ 0.5 (* x x)))
     (if (<= t -4.4e-184)
       t_1
       (if (<= t -2.4e-223)
         t_3
         (if (<= t 5.1e-251)
           (* t (/ (sqrt 2.0) (* (sqrt 2.0) (* l (sqrt (/ 1.0 x))))))
           (if (<= t 1.75e-158)
             (*
              t
              (/
               (sqrt 2.0)
               (fma
                0.5
                (/ (+ t_2 t_2) (* (sqrt 2.0) (* t x)))
                (* t (sqrt 2.0)))))
             (if (<= t 1e+139) t_1 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / fma((l / x), l, ((2.0 * (t * (t + (t / x)))) + (l * (l / x))))));
	double t_2 = fma(2.0, (t * t), (l * l));
	double t_3 = (1.0 / x) + -1.0;
	double tmp;
	if (t <= -1e+146) {
		tmp = t_3 - (0.5 / (x * x));
	} else if (t <= -4.4e-184) {
		tmp = t_1;
	} else if (t <= -2.4e-223) {
		tmp = t_3;
	} else if (t <= 5.1e-251) {
		tmp = t * (sqrt(2.0) / (sqrt(2.0) * (l * sqrt((1.0 / x)))));
	} else if (t <= 1.75e-158) {
		tmp = t * (sqrt(2.0) / fma(0.5, ((t_2 + t_2) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 1e+139) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / fma(Float64(l / x), l, Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(l * Float64(l / x)))))))
	t_2 = fma(2.0, Float64(t * t), Float64(l * l))
	t_3 = Float64(Float64(1.0 / x) + -1.0)
	tmp = 0.0
	if (t <= -1e+146)
		tmp = Float64(t_3 - Float64(0.5 / Float64(x * x)));
	elseif (t <= -4.4e-184)
		tmp = t_1;
	elseif (t <= -2.4e-223)
		tmp = t_3;
	elseif (t <= 5.1e-251)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(sqrt(2.0) * Float64(l * sqrt(Float64(1.0 / x))))));
	elseif (t <= 1.75e-158)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(t_2 + t_2) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 1e+139)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(l / x), $MachinePrecision] * l + N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[t, -1e+146], N[(t$95$3 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.4e-184], t$95$1, If[LessEqual[t, -2.4e-223], t$95$3, If[LessEqual[t, 5.1e-251], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.75e-158], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(t$95$2 + t$95$2), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+139], t$95$1, 1.0]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -9.99999999999999934e145

   1. Initial program 2.5%

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

      1. associate-/l* 2.5%

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

      2. fma-neg 2.5%

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

      3. remove-double-neg 2.5%

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

      4. fma-neg 2.5%

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

      5. sub-neg 2.5%

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

      6. metadata-eval 2.5%

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

      7. remove-double-neg 2.5%

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

      8. fma-def 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-rgt-neg-in 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. metadata-eval 97.8%

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

      6. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around inf 97.8%

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

      1. associate--r+ 97.8%

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

      2. sub-neg 97.8%

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

      3. metadata-eval 97.8%

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

      4. associate-*r/ 97.8%

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

      5. metadata-eval 97.8%

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

      6. unpow2 97.8%

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

   9. Simplified 97.8%

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

   if -9.99999999999999934e145 < t < -4.39999999999999984e-184 or 1.75000000000000006e-158 < t < 1.00000000000000003e139

   1. Initial program 51.3%

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

      1. associate-*l/ 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around inf 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%

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

   7. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

      3. unpow2 79.6%

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

      4. distribute-rgt-neg-in 79.6%

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

   9. Simplified 79.6%

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

       1. *-un-lft-identity 79.6%

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

       2. associate-/l* 79.6%

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

       3. +-commutative 79.6%

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

       4. associate-/l* 79.6%

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

       5. fma-udef 79.6%

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

       6. associate-/r/ 79.6%

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

   11. Applied egg-rr 79.6%

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

       1. *-lft-identity 79.6%

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

       2. associate-/r/ 79.6%

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

       3. fma-def 79.6%

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

       4. +-commutative 79.6%

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

       5. distribute-rgt-out 79.6%

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

       6. associate-/l* 91.7%

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

   13. Simplified 91.7%

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

       1. sqrt-undiv 92.0%

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

       2. fma-def 91.9%

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

       3. +-commutative 91.9%

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

       4. associate-/r/ 92.0%

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

   15. Applied egg-rr 92.0%

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

   if -4.39999999999999984e-184 < t < -2.39999999999999985e-223

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if -2.39999999999999985e-223 < t < 5.10000000000000006e-251

   1. Initial program 5.6%

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

      1. associate-*l/ 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in x around inf 62.8%

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

      1. associate--l+ 62.8%

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

      2. unpow2 62.8%

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

      3. distribute-lft-out 62.8%

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

      4. unpow2 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*r/ 62.8%

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

      7. mul-1-neg 62.8%

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

      8. unpow2 62.8%

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

      9. +-commutative 62.8%

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

      10. unpow2 62.8%

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

      11. fma-udef 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in l around inf 61.3%

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

      1. associate-*l* 61.3%

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

   9. Simplified 61.3%

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

   if 5.10000000000000006e-251 < t < 1.75000000000000006e-158

   1. Initial program 7.6%

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

      1. associate-*l/ 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in x around inf 82.5%

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

      1. fma-def 82.5%

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

   6. Simplified 82.5%

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

   if 1.00000000000000003e139 < t

   1. Initial program 10.7%

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

      1. associate-*l/ 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in x around inf 94.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-184}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) + \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 10^{+139}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \frac{1}{x} + -1\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+145}:\\ \;\;\;\;t_2 - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(-0.5, \frac{\left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right) - \ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (fma (/ l x) l (+ (* 2.0 (* t (+ t (/ t x)))) (* l (/ l x))))))))
        (t_2 (+ (/ 1.0 x) -1.0)))
   (if (<= t -8.2e+145)
     (- t_2 (/ 0.5 (* x x)))
     (if (<= t -5.8e-189)
       t_1
       (if (<= t -4.9e-222)
         t_2
         (if (<= t 2.9e-251)
           (* t (/ (sqrt 2.0) (* (sqrt 2.0) (* l (sqrt (/ 1.0 x))))))
           (if (<= t 5.5e-158)
             (*
              t
              (/
               (sqrt 2.0)
               (fma
                -0.5
                (/
                 (- (- (* (* t t) -2.0) (* l l)) (* l l))
                 (* (sqrt 2.0) (* t x)))
                (* t (sqrt 2.0)))))
             (if (<= t 6.4e+139) t_1 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / fma((l / x), l, ((2.0 * (t * (t + (t / x)))) + (l * (l / x))))));
	double t_2 = (1.0 / x) + -1.0;
	double tmp;
	if (t <= -8.2e+145) {
		tmp = t_2 - (0.5 / (x * x));
	} else if (t <= -5.8e-189) {
		tmp = t_1;
	} else if (t <= -4.9e-222) {
		tmp = t_2;
	} else if (t <= 2.9e-251) {
		tmp = t * (sqrt(2.0) / (sqrt(2.0) * (l * sqrt((1.0 / x)))));
	} else if (t <= 5.5e-158) {
		tmp = t * (sqrt(2.0) / fma(-0.5, (((((t * t) * -2.0) - (l * l)) - (l * l)) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 6.4e+139) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / fma(Float64(l / x), l, Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(l * Float64(l / x)))))))
	t_2 = Float64(Float64(1.0 / x) + -1.0)
	tmp = 0.0
	if (t <= -8.2e+145)
		tmp = Float64(t_2 - Float64(0.5 / Float64(x * x)));
	elseif (t <= -5.8e-189)
		tmp = t_1;
	elseif (t <= -4.9e-222)
		tmp = t_2;
	elseif (t <= 2.9e-251)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(sqrt(2.0) * Float64(l * sqrt(Float64(1.0 / x))))));
	elseif (t <= 5.5e-158)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(-0.5, Float64(Float64(Float64(Float64(Float64(t * t) * -2.0) - Float64(l * l)) - Float64(l * l)) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 6.4e+139)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(l / x), $MachinePrecision] * l + N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[t, -8.2e+145], N[(t$95$2 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5.8e-189], t$95$1, If[LessEqual[t, -4.9e-222], t$95$2, If[LessEqual[t, 2.9e-251], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e-158], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(-0.5 * N[(N[(N[(N[(N[(t * t), $MachinePrecision] * -2.0), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.4e+139], t$95$1, 1.0]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -8.2000000000000003e145

   1. Initial program 2.5%

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

      1. associate-/l* 2.5%

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

      2. fma-neg 2.5%

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

      3. remove-double-neg 2.5%

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

      4. fma-neg 2.5%

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

      5. sub-neg 2.5%

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

      6. metadata-eval 2.5%

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

      7. remove-double-neg 2.5%

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

      8. fma-def 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-rgt-neg-in 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. metadata-eval 97.8%

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

      6. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around inf 97.8%

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

      1. associate--r+ 97.8%

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

      2. sub-neg 97.8%

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

      3. metadata-eval 97.8%

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

      4. associate-*r/ 97.8%

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

      5. metadata-eval 97.8%

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

      6. unpow2 97.8%

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

   9. Simplified 97.8%

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

   if -8.2000000000000003e145 < t < -5.8e-189 or 5.50000000000000025e-158 < t < 6.4000000000000002e139

   1. Initial program 51.3%

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

      1. associate-*l/ 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around inf 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%

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

   7. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

      3. unpow2 79.6%

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

      4. distribute-rgt-neg-in 79.6%

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

   9. Simplified 79.6%

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

       1. *-un-lft-identity 79.6%

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

       2. associate-/l* 79.6%

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

       3. +-commutative 79.6%

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

       4. associate-/l* 79.6%

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

       5. fma-udef 79.6%

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

       6. associate-/r/ 79.6%

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

   11. Applied egg-rr 79.6%

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

       1. *-lft-identity 79.6%

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

       2. associate-/r/ 79.6%

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

       3. fma-def 79.6%

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

       4. +-commutative 79.6%

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

       5. distribute-rgt-out 79.6%

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

       6. associate-/l* 91.7%

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

   13. Simplified 91.7%

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

       1. sqrt-undiv 92.0%

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

       2. fma-def 91.9%

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

       3. +-commutative 91.9%

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

       4. associate-/r/ 92.0%

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

   15. Applied egg-rr 92.0%

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

   if -5.8e-189 < t < -4.9e-222

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if -4.9e-222 < t < 2.9000000000000001e-251

   1. Initial program 5.6%

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

      1. associate-*l/ 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in x around inf 62.8%

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

      1. associate--l+ 62.8%

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

      2. unpow2 62.8%

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

      3. distribute-lft-out 62.8%

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

      4. unpow2 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*r/ 62.8%

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

      7. mul-1-neg 62.8%

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

      8. unpow2 62.8%

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

      9. +-commutative 62.8%

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

      10. unpow2 62.8%

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

      11. fma-udef 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in l around inf 61.3%

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

      1. associate-*l* 61.3%

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

   9. Simplified 61.3%

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

   if 2.9000000000000001e-251 < t < 5.50000000000000025e-158

   1. Initial program 7.6%

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

      1. associate-*l/ 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in x around inf 22.4%

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

      1. associate--l+ 22.4%

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

      2. unpow2 22.4%

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

      3. distribute-lft-out 22.4%

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

      4. unpow2 22.4%

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

      5. unpow2 22.4%

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

      6. associate-*r/ 22.4%

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

      7. mul-1-neg 22.4%

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

      8. unpow2 22.4%

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

      9. +-commutative 22.4%

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

      10. unpow2 22.4%

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

      11. fma-udef 22.4%

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

   6. Simplified 22.4%

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

   7. Taylor expanded in t around 0 21.9%

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

      1. associate-*r/ 21.9%

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

      2. neg-mul-1 21.9%

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

      3. unpow2 21.9%

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

      4. distribute-rgt-neg-in 21.9%

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

   9. Simplified 21.9%

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

       1. *-un-lft-identity 21.9%

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

       2. associate-/l* 21.9%

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

       3. +-commutative 21.9%

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

       4. associate-/l* 21.9%

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

       5. fma-udef 21.9%

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

       6. associate-/r/ 21.9%

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

   11. Applied egg-rr 21.9%

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

       1. *-lft-identity 21.9%

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

       2. associate-/r/ 21.9%

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

       3. fma-def 21.9%

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

       4. +-commutative 21.9%

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

       5. distribute-rgt-out 21.9%

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

       6. associate-/l* 21.9%

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

   13. Simplified 21.9%

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

   14. Taylor expanded in x around -inf 79.8%

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

       1. fma-def 79.8%

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

       2. mul-1-neg 79.8%

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

       3. sub-neg 79.8%

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

       4. *-commutative 79.8%

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

       5. unpow2 79.8%

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

       6. unpow2 79.8%

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

       7. unpow2 79.8%

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

   16. Simplified 79.8%

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

   if 6.4000000000000002e139 < t

   1. Initial program 10.7%

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

      1. associate-*l/ 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in x around inf 94.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(-0.5, \frac{\left(\left(t \cdot t\right) \cdot -2 - \ell \cdot \ell\right) - \ell \cdot \ell}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{+139}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \frac{1}{x} + -1\\ t_3 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+146}:\\ \;\;\;\;t_2 - t_3\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-157}:\\ \;\;\;\;1 + \left(t_3 + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (fma (/ l x) l (+ (* 2.0 (* t (+ t (/ t x)))) (* l (/ l x))))))))
        (t_2 (+ (/ 1.0 x) -1.0))
        (t_3 (/ 0.5 (* x x))))
   (if (<= t -1e+146)
     (- t_2 t_3)
     (if (<= t -4.7e-186)
       t_1
       (if (<= t -6.5e-222)
         t_2
         (if (<= t 6.1e-251)
           (* t (/ (sqrt 2.0) (* (sqrt 2.0) (* l (sqrt (/ 1.0 x))))))
           (if (<= t 1.7e-157)
             (+ 1.0 (+ t_3 (/ -1.0 x)))
             (if (<= t 2.1e+139) t_1 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / fma((l / x), l, ((2.0 * (t * (t + (t / x)))) + (l * (l / x))))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -1e+146) {
		tmp = t_2 - t_3;
	} else if (t <= -4.7e-186) {
		tmp = t_1;
	} else if (t <= -6.5e-222) {
		tmp = t_2;
	} else if (t <= 6.1e-251) {
		tmp = t * (sqrt(2.0) / (sqrt(2.0) * (l * sqrt((1.0 / x)))));
	} else if (t <= 1.7e-157) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 2.1e+139) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / fma(Float64(l / x), l, Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(l * Float64(l / x)))))))
	t_2 = Float64(Float64(1.0 / x) + -1.0)
	t_3 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -1e+146)
		tmp = Float64(t_2 - t_3);
	elseif (t <= -4.7e-186)
		tmp = t_1;
	elseif (t <= -6.5e-222)
		tmp = t_2;
	elseif (t <= 6.1e-251)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(sqrt(2.0) * Float64(l * sqrt(Float64(1.0 / x))))));
	elseif (t <= 1.7e-157)
		tmp = Float64(1.0 + Float64(t_3 + Float64(-1.0 / x)));
	elseif (t <= 2.1e+139)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(l / x), $MachinePrecision] * l + N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+146], N[(t$95$2 - t$95$3), $MachinePrecision], If[LessEqual[t, -4.7e-186], t$95$1, If[LessEqual[t, -6.5e-222], t$95$2, If[LessEqual[t, 6.1e-251], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(l * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.7e-157], N[(1.0 + N[(t$95$3 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.1e+139], t$95$1, 1.0]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -9.99999999999999934e145

   1. Initial program 2.5%

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

      1. associate-/l* 2.5%

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

      2. fma-neg 2.5%

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

      3. remove-double-neg 2.5%

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

      4. fma-neg 2.5%

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

      5. sub-neg 2.5%

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

      6. metadata-eval 2.5%

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

      7. remove-double-neg 2.5%

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

      8. fma-def 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-rgt-neg-in 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. metadata-eval 97.8%

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

      6. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around inf 97.8%

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

      1. associate--r+ 97.8%

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

      2. sub-neg 97.8%

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

      3. metadata-eval 97.8%

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

      4. associate-*r/ 97.8%

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

      5. metadata-eval 97.8%

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

      6. unpow2 97.8%

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

   9. Simplified 97.8%

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

   if -9.99999999999999934e145 < t < -4.6999999999999997e-186 or 1.69999999999999989e-157 < t < 2.0999999999999999e139

   1. Initial program 51.3%

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

      1. associate-*l/ 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around inf 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%

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

   7. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

      3. unpow2 79.6%

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

      4. distribute-rgt-neg-in 79.6%

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

   9. Simplified 79.6%

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

       1. *-un-lft-identity 79.6%

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

       2. associate-/l* 79.6%

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

       3. +-commutative 79.6%

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

       4. associate-/l* 79.6%

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

       5. fma-udef 79.6%

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

       6. associate-/r/ 79.6%

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

   11. Applied egg-rr 79.6%

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

       1. *-lft-identity 79.6%

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

       2. associate-/r/ 79.6%

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

       3. fma-def 79.6%

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

       4. +-commutative 79.6%

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

       5. distribute-rgt-out 79.6%

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

       6. associate-/l* 91.7%

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

   13. Simplified 91.7%

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

       1. sqrt-undiv 92.0%

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

       2. fma-def 91.9%

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

       3. +-commutative 91.9%

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

       4. associate-/r/ 92.0%

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

   15. Applied egg-rr 92.0%

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

   if -4.6999999999999997e-186 < t < -6.5000000000000005e-222

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if -6.5000000000000005e-222 < t < 6.1000000000000003e-251

   1. Initial program 5.6%

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

      1. associate-*l/ 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in x around inf 62.8%

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

      1. associate--l+ 62.8%

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

      2. unpow2 62.8%

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

      3. distribute-lft-out 62.8%

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

      4. unpow2 62.8%

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

      5. unpow2 62.8%

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

      6. associate-*r/ 62.8%

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

      7. mul-1-neg 62.8%

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

      8. unpow2 62.8%

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

      9. +-commutative 62.8%

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

      10. unpow2 62.8%

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

      11. fma-udef 62.8%

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

   6. Simplified 62.8%

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

   7. Taylor expanded in l around inf 61.3%

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

      1. associate-*l* 61.3%

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

   9. Simplified 61.3%

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

   if 6.1000000000000003e-251 < t < 1.69999999999999989e-157

   1. Initial program 7.6%

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

      1. associate-*l/ 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in t around inf 73.7%

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

   5. Taylor expanded in x around inf 74.0%

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

      1. associate--l+ 74.1%

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

      2. associate-*r/ 74.1%

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

      3. metadata-eval 74.1%

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

      4. unpow2 74.1%

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

   7. Simplified 74.1%

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

   if 2.0999999999999999e139 < t

   1. Initial program 10.7%

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

      1. associate-*l/ 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in x around inf 94.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+146}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -4.7 \cdot 10^{-186}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{-251}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2} \cdot \left(\ell \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-157}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+139}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \frac{1}{x} + -1\\ t_3 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+146}:\\ \;\;\;\;t_2 - t_3\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-252}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\frac{\ell}{\sqrt{0.5}}} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-158}:\\ \;\;\;\;1 + \left(t_3 + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (fma (/ l x) l (+ (* 2.0 (* t (+ t (/ t x)))) (* l (/ l x))))))))
        (t_2 (+ (/ 1.0 x) -1.0))
        (t_3 (/ 0.5 (* x x))))
   (if (<= t -5e+146)
     (- t_2 t_3)
     (if (<= t -3.3e-189)
       t_1
       (if (<= t -4e-222)
         t_2
         (if (<= t 2.5e-252)
           (* t (* (/ (sqrt 2.0) (/ l (sqrt 0.5))) (sqrt x)))
           (if (<= t 6e-158)
             (+ 1.0 (+ t_3 (/ -1.0 x)))
             (if (<= t 1e+140) t_1 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / fma((l / x), l, ((2.0 * (t * (t + (t / x)))) + (l * (l / x))))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -5e+146) {
		tmp = t_2 - t_3;
	} else if (t <= -3.3e-189) {
		tmp = t_1;
	} else if (t <= -4e-222) {
		tmp = t_2;
	} else if (t <= 2.5e-252) {
		tmp = t * ((sqrt(2.0) / (l / sqrt(0.5))) * sqrt(x));
	} else if (t <= 6e-158) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 1e+140) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / fma(Float64(l / x), l, Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(l * Float64(l / x)))))))
	t_2 = Float64(Float64(1.0 / x) + -1.0)
	t_3 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -5e+146)
		tmp = Float64(t_2 - t_3);
	elseif (t <= -3.3e-189)
		tmp = t_1;
	elseif (t <= -4e-222)
		tmp = t_2;
	elseif (t <= 2.5e-252)
		tmp = Float64(t * Float64(Float64(sqrt(2.0) / Float64(l / sqrt(0.5))) * sqrt(x)));
	elseif (t <= 6e-158)
		tmp = Float64(1.0 + Float64(t_3 + Float64(-1.0 / x)));
	elseif (t <= 1e+140)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(l / x), $MachinePrecision] * l + N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e+146], N[(t$95$2 - t$95$3), $MachinePrecision], If[LessEqual[t, -3.3e-189], t$95$1, If[LessEqual[t, -4e-222], t$95$2, If[LessEqual[t, 2.5e-252], N[(t * N[(N[(N[Sqrt[2.0], $MachinePrecision] / N[(l / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-158], N[(1.0 + N[(t$95$3 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+140], t$95$1, 1.0]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.9999999999999999e146

   1. Initial program 2.5%

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

      1. associate-/l* 2.5%

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

      2. fma-neg 2.5%

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

      3. remove-double-neg 2.5%

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

      4. fma-neg 2.5%

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

      5. sub-neg 2.5%

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

      6. metadata-eval 2.5%

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

      7. remove-double-neg 2.5%

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

      8. fma-def 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-rgt-neg-in 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. metadata-eval 97.8%

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

      6. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around inf 97.8%

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

      1. associate--r+ 97.8%

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

      2. sub-neg 97.8%

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

      3. metadata-eval 97.8%

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

      4. associate-*r/ 97.8%

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

      5. metadata-eval 97.8%

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

      6. unpow2 97.8%

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

   9. Simplified 97.8%

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

   if -4.9999999999999999e146 < t < -3.3000000000000001e-189 or 6e-158 < t < 1.00000000000000006e140

   1. Initial program 51.3%

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

      1. associate-*l/ 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around inf 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%

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

   7. Taylor expanded in t around 0 79.6%

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

      1. associate-*r/ 79.6%

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

      2. neg-mul-1 79.6%

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

      3. unpow2 79.6%

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

      4. distribute-rgt-neg-in 79.6%

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

   9. Simplified 79.6%

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

       1. *-un-lft-identity 79.6%

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

       2. associate-/l* 79.6%

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

       3. +-commutative 79.6%

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

       4. associate-/l* 79.6%

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

       5. fma-udef 79.6%

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

       6. associate-/r/ 79.6%

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

   11. Applied egg-rr 79.6%

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

       1. *-lft-identity 79.6%

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

       2. associate-/r/ 79.6%

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

       3. fma-def 79.6%

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

       4. +-commutative 79.6%

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

       5. distribute-rgt-out 79.6%

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

       6. associate-/l* 91.7%

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

   13. Simplified 91.7%

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

       1. sqrt-undiv 92.0%

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

       2. fma-def 91.9%

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

       3. +-commutative 91.9%

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

       4. associate-/r/ 92.0%

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

   15. Applied egg-rr 92.0%

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

   if -3.3000000000000001e-189 < t < -4.00000000000000019e-222

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if -4.00000000000000019e-222 < t < 2.50000000000000004e-252

   1. Initial program 5.6%

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

      1. associate-*l/ 5.6%

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

   3. Simplified 5.6%

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

      1. expm1-log1p-u 5.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 5.6%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 5.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 5.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 5.5%

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

      3. metadata-eval 5.5%

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

      4. sub-neg 5.5%

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

      5. associate-/l* 5.4%

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

      6. sub-neg 5.4%

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

      7. metadata-eval 5.4%

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

      8. +-commutative 5.4%

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

   7. Simplified 5.4%

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

   8. Taylor expanded in x around inf 62.8%

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

      1. distribute-lft-in 62.8%

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

      2. associate--l+ 62.8%

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

      3. unpow2 62.8%

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

      4. +-commutative 62.8%

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

      5. unpow2 62.8%

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

      6. unpow2 62.8%

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

      7. fma-def 62.8%

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

      8. associate-/l* 62.8%

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

      9. associate-*r/ 62.8%

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

   10. Simplified 62.8%

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

   11. Taylor expanded in l around inf 61.0%

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

       1. associate-/l* 61.0%

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

   13. Simplified 61.0%

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

   if 2.50000000000000004e-252 < t < 6e-158

   1. Initial program 7.6%

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

      1. associate-*l/ 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in t around inf 73.7%

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

   5. Taylor expanded in x around inf 74.0%

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

      1. associate--l+ 74.1%

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

      2. associate-*r/ 74.1%

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

      3. metadata-eval 74.1%

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

      4. unpow2 74.1%

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

   7. Simplified 74.1%

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

   if 1.00000000000000006e140 < t

   1. Initial program 10.7%

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

      1. associate-*l/ 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in x around inf 94.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+146}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-189}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-252}:\\ \;\;\;\;t \cdot \left(\frac{\sqrt{2}}{\frac{\ell}{\sqrt{0.5}}} \cdot \sqrt{x}\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-158}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 10^{+140}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ t_2 := \frac{1}{x} + -1\\ t_3 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+145}:\\ \;\;\;\;t_2 - t_3\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;1 + \left(t_3 + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (fma (/ l x) l (+ (* 2.0 (* t (+ t (/ t x)))) (* l (/ l x))))))))
        (t_2 (+ (/ 1.0 x) -1.0))
        (t_3 (/ 0.5 (* x x))))
   (if (<= t -4e+145)
     (- t_2 t_3)
     (if (<= t -1.35e-187)
       t_1
       (if (<= t -5.8e-223)
         t_2
         (if (<= t 8.2e-256)
           t_1
           (if (<= t 1.05e-157)
             (+ 1.0 (+ t_3 (/ -1.0 x)))
             (if (<= t 6.5e+139) t_1 1.0))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / fma((l / x), l, ((2.0 * (t * (t + (t / x)))) + (l * (l / x))))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -4e+145) {
		tmp = t_2 - t_3;
	} else if (t <= -1.35e-187) {
		tmp = t_1;
	} else if (t <= -5.8e-223) {
		tmp = t_2;
	} else if (t <= 8.2e-256) {
		tmp = t_1;
	} else if (t <= 1.05e-157) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 6.5e+139) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / fma(Float64(l / x), l, Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(l * Float64(l / x)))))))
	t_2 = Float64(Float64(1.0 / x) + -1.0)
	t_3 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -4e+145)
		tmp = Float64(t_2 - t_3);
	elseif (t <= -1.35e-187)
		tmp = t_1;
	elseif (t <= -5.8e-223)
		tmp = t_2;
	elseif (t <= 8.2e-256)
		tmp = t_1;
	elseif (t <= 1.05e-157)
		tmp = Float64(1.0 + Float64(t_3 + Float64(-1.0 / x)));
	elseif (t <= 6.5e+139)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(l / x), $MachinePrecision] * l + N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4e+145], N[(t$95$2 - t$95$3), $MachinePrecision], If[LessEqual[t, -1.35e-187], t$95$1, If[LessEqual[t, -5.8e-223], t$95$2, If[LessEqual[t, 8.2e-256], t$95$1, If[LessEqual[t, 1.05e-157], N[(1.0 + N[(t$95$3 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.5e+139], t$95$1, 1.0]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4e145

   1. Initial program 2.5%

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

      1. associate-/l* 2.5%

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

      2. fma-neg 2.5%

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

      3. remove-double-neg 2.5%

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

      4. fma-neg 2.5%

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

      5. sub-neg 2.5%

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

      6. metadata-eval 2.5%

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

      7. remove-double-neg 2.5%

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

      8. fma-def 2.5%

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

   3. Simplified 2.5%

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

   4. Taylor expanded in t around -inf 97.8%

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

      1. mul-1-neg 97.8%

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

      2. distribute-rgt-neg-in 97.8%

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

      3. +-commutative 97.8%

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

      4. sub-neg 97.8%

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

      5. metadata-eval 97.8%

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

      6. +-commutative 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around inf 97.8%

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

      1. associate--r+ 97.8%

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

      2. sub-neg 97.8%

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

      3. metadata-eval 97.8%

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

      4. associate-*r/ 97.8%

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

      5. metadata-eval 97.8%

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

      6. unpow2 97.8%

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

   9. Simplified 97.8%

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

   if -4e145 < t < -1.35e-187 or -5.8000000000000001e-223 < t < 8.2e-256 or 1.05e-157 < t < 6.5000000000000003e139

   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-*l/ 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in x around inf 77.8%

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

      1. associate--l+ 77.8%

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

      2. unpow2 77.8%

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

      3. distribute-lft-out 77.8%

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

      4. unpow2 77.8%

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

      5. unpow2 77.8%

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

      6. associate-*r/ 77.8%

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

      7. mul-1-neg 77.8%

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

      8. unpow2 77.8%

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

      9. +-commutative 77.8%

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

      10. unpow2 77.8%

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

      11. fma-udef 77.8%

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

   6. Simplified 77.8%

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

   7. Taylor expanded in t around 0 77.1%

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

      1. associate-*r/ 77.1%

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

      2. neg-mul-1 77.1%

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

      3. unpow2 77.1%

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

      4. distribute-rgt-neg-in 77.1%

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

   9. Simplified 77.1%

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

       1. *-un-lft-identity 77.1%

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

       2. associate-/l* 77.1%

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

       3. +-commutative 77.1%

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

       4. associate-/l* 77.1%

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

       5. fma-udef 77.1%

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

       6. associate-/r/ 77.1%

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

   11. Applied egg-rr 77.1%

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

       1. *-lft-identity 77.1%

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

       2. associate-/r/ 77.1%

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

       3. fma-def 77.1%

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

       4. +-commutative 77.1%

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

       5. distribute-rgt-out 77.1%

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

       6. associate-/l* 87.7%

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

   13. Simplified 87.7%

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

       1. sqrt-undiv 87.9%

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

       2. fma-def 87.9%

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

       3. +-commutative 87.9%

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

       4. associate-/r/ 87.9%

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

   15. Applied egg-rr 87.9%

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

   if -1.35e-187 < t < -5.8000000000000001e-223

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 8.2e-256 < t < 1.05e-157

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in t around inf 68.9%

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

   5. Taylor expanded in x around inf 69.2%

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

      1. associate--l+ 69.3%

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

      2. associate-*r/ 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow2 69.3%

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

   7. Simplified 69.3%

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

   if 6.5000000000000003e139 < t

   1. Initial program 10.7%

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

      1. associate-*l/ 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in x around inf 94.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 96.1%

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

      2. metadata-eval 96.1%

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

      3. metadata-eval 96.1%

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

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+145}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-187}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-223}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-256}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-157}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+139}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(\frac{\ell}{x}, \ell, 2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 81.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell \cdot \ell}{x}\right) + 2 \cdot \frac{t \cdot t}{x}}}\\ t_2 := \frac{1}{x} + -1\\ t_3 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+108}:\\ \;\;\;\;t_2 - t_3\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-158}:\\ \;\;\;\;1 + \left(t_3 + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (+
             (* 2.0 (+ (* t (+ t (/ t x))) (/ (* l l) x)))
             (* 2.0 (/ (* t t) x)))))))
        (t_2 (+ (/ 1.0 x) -1.0))
        (t_3 (/ 0.5 (* x x))))
   (if (<= t -8.8e+108)
     (- t_2 t_3)
     (if (<= t -1.7e-187)
       t_1
       (if (<= t -4.9e-222)
         t_2
         (if (<= t 1.08e-253)
           t_1
           (if (<= t 5e-158)
             (+ 1.0 (+ t_3 (/ -1.0 x)))
             (if (<= t 5.6e+49) t_1 (+ 1.0 (/ -1.0 x))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / ((2.0 * ((t * (t + (t / x))) + ((l * l) / x))) + (2.0 * ((t * t) / x)))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -8.8e+108) {
		tmp = t_2 - t_3;
	} else if (t <= -1.7e-187) {
		tmp = t_1;
	} else if (t <= -4.9e-222) {
		tmp = t_2;
	} else if (t <= 1.08e-253) {
		tmp = t_1;
	} else if (t <= 5e-158) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 5.6e+49) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = t * sqrt((2.0d0 / ((2.0d0 * ((t * (t + (t / x))) + ((l * l) / x))) + (2.0d0 * ((t * t) / x)))))
    t_2 = (1.0d0 / x) + (-1.0d0)
    t_3 = 0.5d0 / (x * x)
    if (t <= (-8.8d+108)) then
        tmp = t_2 - t_3
    else if (t <= (-1.7d-187)) then
        tmp = t_1
    else if (t <= (-4.9d-222)) then
        tmp = t_2
    else if (t <= 1.08d-253) then
        tmp = t_1
    else if (t <= 5d-158) then
        tmp = 1.0d0 + (t_3 + ((-1.0d0) / x))
    else if (t <= 5.6d+49) then
        tmp = t_1
    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 t_1 = t * Math.sqrt((2.0 / ((2.0 * ((t * (t + (t / x))) + ((l * l) / x))) + (2.0 * ((t * t) / x)))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -8.8e+108) {
		tmp = t_2 - t_3;
	} else if (t <= -1.7e-187) {
		tmp = t_1;
	} else if (t <= -4.9e-222) {
		tmp = t_2;
	} else if (t <= 1.08e-253) {
		tmp = t_1;
	} else if (t <= 5e-158) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 5.6e+49) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * math.sqrt((2.0 / ((2.0 * ((t * (t + (t / x))) + ((l * l) / x))) + (2.0 * ((t * t) / x)))))
	t_2 = (1.0 / x) + -1.0
	t_3 = 0.5 / (x * x)
	tmp = 0
	if t <= -8.8e+108:
		tmp = t_2 - t_3
	elif t <= -1.7e-187:
		tmp = t_1
	elif t <= -4.9e-222:
		tmp = t_2
	elif t <= 1.08e-253:
		tmp = t_1
	elif t <= 5e-158:
		tmp = 1.0 + (t_3 + (-1.0 / x))
	elif t <= 5.6e+49:
		tmp = t_1
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(Float64(t * Float64(t + Float64(t / x))) + Float64(Float64(l * l) / x))) + Float64(2.0 * Float64(Float64(t * t) / x))))))
	t_2 = Float64(Float64(1.0 / x) + -1.0)
	t_3 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -8.8e+108)
		tmp = Float64(t_2 - t_3);
	elseif (t <= -1.7e-187)
		tmp = t_1;
	elseif (t <= -4.9e-222)
		tmp = t_2;
	elseif (t <= 1.08e-253)
		tmp = t_1;
	elseif (t <= 5e-158)
		tmp = Float64(1.0 + Float64(t_3 + Float64(-1.0 / x)));
	elseif (t <= 5.6e+49)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * sqrt((2.0 / ((2.0 * ((t * (t + (t / x))) + ((l * l) / x))) + (2.0 * ((t * t) / x)))));
	t_2 = (1.0 / x) + -1.0;
	t_3 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -8.8e+108)
		tmp = t_2 - t_3;
	elseif (t <= -1.7e-187)
		tmp = t_1;
	elseif (t <= -4.9e-222)
		tmp = t_2;
	elseif (t <= 1.08e-253)
		tmp = t_1;
	elseif (t <= 5e-158)
		tmp = 1.0 + (t_3 + (-1.0 / x));
	elseif (t <= 5.6e+49)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.8e+108], N[(t$95$2 - t$95$3), $MachinePrecision], If[LessEqual[t, -1.7e-187], t$95$1, If[LessEqual[t, -4.9e-222], t$95$2, If[LessEqual[t, 1.08e-253], t$95$1, If[LessEqual[t, 5e-158], N[(1.0 + N[(t$95$3 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e+49], t$95$1, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.8000000000000005e108

   1. Initial program 12.3%

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

      1. associate-/l* 12.3%

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

      2. fma-neg 12.3%

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

      3. remove-double-neg 12.3%

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

      4. fma-neg 12.3%

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

      5. sub-neg 12.3%

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

      6. metadata-eval 12.3%

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

      7. remove-double-neg 12.3%

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

      8. fma-def 12.3%

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

   3. Simplified 12.3%

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

   4. Taylor expanded in t around -inf 95.0%

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

      1. mul-1-neg 95.0%

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

      2. distribute-rgt-neg-in 95.0%

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

      3. +-commutative 95.0%

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

      4. sub-neg 95.0%

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

      5. metadata-eval 95.0%

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

      6. +-commutative 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in x around inf 95.0%

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

      1. associate--r+ 95.0%

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

      2. sub-neg 95.0%

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

      3. metadata-eval 95.0%

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

      4. associate-*r/ 95.0%

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

      5. metadata-eval 95.0%

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

      6. unpow2 95.0%

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

   9. Simplified 95.0%

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

   if -8.8000000000000005e108 < t < -1.7000000000000001e-187 or -4.9e-222 < t < 1.08000000000000003e-253 or 4.99999999999999972e-158 < t < 5.5999999999999996e49

   1. Initial program 41.1%

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

      1. associate-*l/ 41.2%

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

   3. Simplified 41.2%

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

      1. expm1-log1p-u 40.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 22.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 32.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 34.2%

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

      3. metadata-eval 34.2%

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

      4. sub-neg 34.2%

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

      5. associate-/l* 32.9%

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

      6. sub-neg 32.9%

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

      7. metadata-eval 32.9%

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

      8. +-commutative 32.9%

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

   7. Simplified 32.9%

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

   8. Taylor expanded in x around inf 80.9%

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

      1. distribute-lft-in 80.9%

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

      2. associate--l+ 80.9%

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

      3. unpow2 80.9%

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

      4. +-commutative 80.9%

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

      5. unpow2 80.9%

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

      6. unpow2 80.9%

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

      7. fma-def 80.9%

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

      8. associate-/l* 80.9%

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

      9. associate-*r/ 80.9%

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

   10. Simplified 80.9%

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

       1. *-un-lft-identity 80.9%

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

       2. associate-/l* 80.9%

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

       3. associate-/r/ 80.9%

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

   12. Applied egg-rr 80.9%

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

       1. *-lft-identity 80.9%

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

       2. associate-/r/ 80.9%

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

       3. fma-def 80.9%

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

       4. +-commutative 80.9%

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

       5. distribute-rgt-out 80.9%

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

   14. Simplified 80.9%

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

   15. Taylor expanded in l around 0 80.9%

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

       1. cancel-sign-sub-inv 80.9%

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

       2. +-commutative 80.9%

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

       3. +-commutative 80.9%

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

       4. distribute-lft-out 80.9%

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

       5. +-commutative 80.9%

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

       6. *-commutative 80.9%

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

       7. unpow2 80.9%

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

       8. metadata-eval 80.9%

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

       9. unpow2 80.9%

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

   17. Simplified 80.9%

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

   if -1.7000000000000001e-187 < t < -4.9e-222

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 1.08000000000000003e-253 < t < 4.99999999999999972e-158

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in t around inf 68.9%

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

   5. Taylor expanded in x around inf 69.2%

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

      1. associate--l+ 69.3%

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

      2. associate-*r/ 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow2 69.3%

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

   7. Simplified 69.3%

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

   if 5.5999999999999996e49 < t

   1. Initial program 18.8%

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

      1. associate-*l/ 18.8%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in t around inf 89.1%

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

   5. Taylor expanded in x around inf 89.5%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+108}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-187}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell \cdot \ell}{x}\right) + 2 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{elif}\;t \leq -4.9 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-253}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell \cdot \ell}{x}\right) + 2 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-158}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+49}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right) + \frac{\ell \cdot \ell}{x}\right) + 2 \cdot \frac{t \cdot t}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 7: 81.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \frac{\ell \cdot \ell}{x}\right)}}\\ t_2 := \frac{1}{x} + -1\\ t_3 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{+108}:\\ \;\;\;\;t_2 - t_3\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-255}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-158}:\\ \;\;\;\;1 + \left(t_3 + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (*
          t
          (sqrt
           (/
            2.0
            (+ (* l (/ l x)) (+ (* 2.0 (* t (+ t (/ t x)))) (/ (* l l) x)))))))
        (t_2 (+ (/ 1.0 x) -1.0))
        (t_3 (/ 0.5 (* x x))))
   (if (<= t -8.8e+108)
     (- t_2 t_3)
     (if (<= t -8.8e-187)
       t_1
       (if (<= t -6.2e-222)
         t_2
         (if (<= t 4.5e-255)
           t_1
           (if (<= t 5e-158)
             (+ 1.0 (+ t_3 (/ -1.0 x)))
             (if (<= t 5.8e+49) t_1 (+ 1.0 (/ -1.0 x))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt((2.0 / ((l * (l / x)) + ((2.0 * (t * (t + (t / x)))) + ((l * l) / x)))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -8.8e+108) {
		tmp = t_2 - t_3;
	} else if (t <= -8.8e-187) {
		tmp = t_1;
	} else if (t <= -6.2e-222) {
		tmp = t_2;
	} else if (t <= 4.5e-255) {
		tmp = t_1;
	} else if (t <= 5e-158) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 5.8e+49) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = t * sqrt((2.0d0 / ((l * (l / x)) + ((2.0d0 * (t * (t + (t / x)))) + ((l * l) / x)))))
    t_2 = (1.0d0 / x) + (-1.0d0)
    t_3 = 0.5d0 / (x * x)
    if (t <= (-8.8d+108)) then
        tmp = t_2 - t_3
    else if (t <= (-8.8d-187)) then
        tmp = t_1
    else if (t <= (-6.2d-222)) then
        tmp = t_2
    else if (t <= 4.5d-255) then
        tmp = t_1
    else if (t <= 5d-158) then
        tmp = 1.0d0 + (t_3 + ((-1.0d0) / x))
    else if (t <= 5.8d+49) then
        tmp = t_1
    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 t_1 = t * Math.sqrt((2.0 / ((l * (l / x)) + ((2.0 * (t * (t + (t / x)))) + ((l * l) / x)))));
	double t_2 = (1.0 / x) + -1.0;
	double t_3 = 0.5 / (x * x);
	double tmp;
	if (t <= -8.8e+108) {
		tmp = t_2 - t_3;
	} else if (t <= -8.8e-187) {
		tmp = t_1;
	} else if (t <= -6.2e-222) {
		tmp = t_2;
	} else if (t <= 4.5e-255) {
		tmp = t_1;
	} else if (t <= 5e-158) {
		tmp = 1.0 + (t_3 + (-1.0 / x));
	} else if (t <= 5.8e+49) {
		tmp = t_1;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * math.sqrt((2.0 / ((l * (l / x)) + ((2.0 * (t * (t + (t / x)))) + ((l * l) / x)))))
	t_2 = (1.0 / x) + -1.0
	t_3 = 0.5 / (x * x)
	tmp = 0
	if t <= -8.8e+108:
		tmp = t_2 - t_3
	elif t <= -8.8e-187:
		tmp = t_1
	elif t <= -6.2e-222:
		tmp = t_2
	elif t <= 4.5e-255:
		tmp = t_1
	elif t <= 5e-158:
		tmp = 1.0 + (t_3 + (-1.0 / x))
	elif t <= 5.8e+49:
		tmp = t_1
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(Float64(2.0 / Float64(Float64(l * Float64(l / x)) + Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(Float64(l * l) / x))))))
	t_2 = Float64(Float64(1.0 / x) + -1.0)
	t_3 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -8.8e+108)
		tmp = Float64(t_2 - t_3);
	elseif (t <= -8.8e-187)
		tmp = t_1;
	elseif (t <= -6.2e-222)
		tmp = t_2;
	elseif (t <= 4.5e-255)
		tmp = t_1;
	elseif (t <= 5e-158)
		tmp = Float64(1.0 + Float64(t_3 + Float64(-1.0 / x)));
	elseif (t <= 5.8e+49)
		tmp = t_1;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * sqrt((2.0 / ((l * (l / x)) + ((2.0 * (t * (t + (t / x)))) + ((l * l) / x)))));
	t_2 = (1.0 / x) + -1.0;
	t_3 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -8.8e+108)
		tmp = t_2 - t_3;
	elseif (t <= -8.8e-187)
		tmp = t_1;
	elseif (t <= -6.2e-222)
		tmp = t_2;
	elseif (t <= 4.5e-255)
		tmp = t_1;
	elseif (t <= 5e-158)
		tmp = 1.0 + (t_3 + (-1.0 / x));
	elseif (t <= 5.8e+49)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(t * N[Sqrt[N[(2.0 / N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.8e+108], N[(t$95$2 - t$95$3), $MachinePrecision], If[LessEqual[t, -8.8e-187], t$95$1, If[LessEqual[t, -6.2e-222], t$95$2, If[LessEqual[t, 4.5e-255], t$95$1, If[LessEqual[t, 5e-158], N[(1.0 + N[(t$95$3 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+49], t$95$1, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.8000000000000005e108

   1. Initial program 12.3%

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

      1. associate-/l* 12.3%

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

      2. fma-neg 12.3%

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

      3. remove-double-neg 12.3%

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

      4. fma-neg 12.3%

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

      5. sub-neg 12.3%

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

      6. metadata-eval 12.3%

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

      7. remove-double-neg 12.3%

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

      8. fma-def 12.3%

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

   3. Simplified 12.3%

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

   4. Taylor expanded in t around -inf 95.0%

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

      1. mul-1-neg 95.0%

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

      2. distribute-rgt-neg-in 95.0%

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

      3. +-commutative 95.0%

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

      4. sub-neg 95.0%

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

      5. metadata-eval 95.0%

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

      6. +-commutative 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in x around inf 95.0%

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

      1. associate--r+ 95.0%

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

      2. sub-neg 95.0%

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

      3. metadata-eval 95.0%

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

      4. associate-*r/ 95.0%

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

      5. metadata-eval 95.0%

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

      6. unpow2 95.0%

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

   9. Simplified 95.0%

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

   if -8.8000000000000005e108 < t < -8.80000000000000032e-187 or -6.19999999999999959e-222 < t < 4.49999999999999979e-255 or 4.99999999999999972e-158 < t < 5.8e49

   1. Initial program 41.1%

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

      1. associate-*l/ 41.2%

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

   3. Simplified 41.2%

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

      1. expm1-log1p-u 40.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 22.1%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 22.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 32.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 34.2%

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

      3. metadata-eval 34.2%

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

      4. sub-neg 34.2%

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

      5. associate-/l* 32.9%

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

      6. sub-neg 32.9%

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

      7. metadata-eval 32.9%

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

      8. +-commutative 32.9%

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

   7. Simplified 32.9%

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

   8. Taylor expanded in x around inf 80.9%

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

      1. distribute-lft-in 80.9%

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

      2. associate--l+ 80.9%

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

      3. unpow2 80.9%

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

      4. +-commutative 80.9%

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

      5. unpow2 80.9%

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

      6. unpow2 80.9%

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

      7. fma-def 80.9%

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

      8. associate-/l* 80.9%

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

      9. associate-*r/ 80.9%

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

   10. Simplified 80.9%

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

       1. *-un-lft-identity 80.9%

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

       2. associate-/l* 80.9%

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

       3. associate-/r/ 80.9%

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

   12. Applied egg-rr 80.9%

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

       1. *-lft-identity 80.9%

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

       2. associate-/r/ 80.9%

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

       3. fma-def 80.9%

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

       4. +-commutative 80.9%

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

       5. distribute-rgt-out 80.9%

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

   14. Simplified 80.9%

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

   15. Taylor expanded in t around 0 80.3%

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

       1. associate-*r/ 80.1%

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

       2. neg-mul-1 80.1%

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

       3. unpow2 80.1%

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

       4. distribute-rgt-neg-in 80.1%

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

   17. Simplified 80.3%

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

   if -8.80000000000000032e-187 < t < -6.19999999999999959e-222

   1. Initial program 3.0%

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

      1. associate-/l* 3.0%

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

      2. fma-neg 3.0%

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

      3. remove-double-neg 3.0%

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

      4. fma-neg 3.0%

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

      5. sub-neg 3.0%

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

      6. metadata-eval 3.0%

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

      7. remove-double-neg 3.0%

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

      8. fma-def 3.0%

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

   3. Simplified 3.0%

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

   4. Taylor expanded in t around -inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. distribute-rgt-neg-in 74.7%

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

      3. +-commutative 74.7%

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

      4. sub-neg 74.7%

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

      5. metadata-eval 74.7%

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

      6. +-commutative 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around inf 74.7%

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

   if 4.49999999999999979e-255 < t < 4.99999999999999972e-158

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in t around inf 68.9%

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

   5. Taylor expanded in x around inf 69.2%

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

      1. associate--l+ 69.3%

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

      2. associate-*r/ 69.3%

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

      3. metadata-eval 69.3%

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

      4. unpow2 69.3%

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

   7. Simplified 69.3%

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

   if 5.8e49 < t

   1. Initial program 18.8%

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

      1. associate-*l/ 18.8%

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

   3. Simplified 18.8%

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

   4. Taylor expanded in t around inf 89.1%

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

   5. Taylor expanded in x around inf 89.5%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+108}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -8.8 \cdot 10^{-187}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-255}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-158}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+49}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \left(2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \frac{\ell \cdot \ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 8: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-223}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-305}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(\frac{2}{x} + \left(2 + \frac{2}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -1.5e-223)
     (- (+ (/ 1.0 x) -1.0) t_1)
     (if (<= t 2.1e-305)
       (*
        t
        (sqrt
         (/
          2.0
          (+ (* l (/ l x)) (* (* t t) (+ (/ 2.0 x) (+ 2.0 (/ 2.0 x))))))))
       (+ 1.0 (+ t_1 (/ -1.0 x)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1.5e-223) {
		tmp = ((1.0 / x) + -1.0) - t_1;
	} else if (t <= 2.1e-305) {
		tmp = t * sqrt((2.0 / ((l * (l / x)) + ((t * t) * ((2.0 / x) + (2.0 + (2.0 / x)))))));
	} else {
		tmp = 1.0 + (t_1 + (-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) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-1.5d-223)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) - t_1
    else if (t <= 2.1d-305) then
        tmp = t * sqrt((2.0d0 / ((l * (l / x)) + ((t * t) * ((2.0d0 / x) + (2.0d0 + (2.0d0 / x)))))))
    else
        tmp = 1.0d0 + (t_1 + ((-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 t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1.5e-223) {
		tmp = ((1.0 / x) + -1.0) - t_1;
	} else if (t <= 2.1e-305) {
		tmp = t * Math.sqrt((2.0 / ((l * (l / x)) + ((t * t) * ((2.0 / x) + (2.0 + (2.0 / x)))))));
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -1.5e-223:
		tmp = ((1.0 / x) + -1.0) - t_1
	elif t <= 2.1e-305:
		tmp = t * math.sqrt((2.0 / ((l * (l / x)) + ((t * t) * ((2.0 / x) + (2.0 + (2.0 / x)))))))
	else:
		tmp = 1.0 + (t_1 + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -1.5e-223)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - t_1);
	elseif (t <= 2.1e-305)
		tmp = Float64(t * sqrt(Float64(2.0 / Float64(Float64(l * Float64(l / x)) + Float64(Float64(t * t) * Float64(Float64(2.0 / x) + Float64(2.0 + Float64(2.0 / x))))))));
	else
		tmp = Float64(1.0 + Float64(t_1 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -1.5e-223)
		tmp = ((1.0 / x) + -1.0) - t_1;
	elseif (t <= 2.1e-305)
		tmp = t * sqrt((2.0 / ((l * (l / x)) + ((t * t) * ((2.0 / x) + (2.0 + (2.0 / x)))))));
	else
		tmp = 1.0 + (t_1 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e-223], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t, 2.1e-305], N[(t * N[Sqrt[N[(2.0 / N[(N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] * N[(N[(2.0 / x), $MachinePrecision] + N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.49999999999999996e-223

   1. Initial program 32.9%

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

      1. associate-/l* 32.8%

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

      2. fma-neg 32.8%

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

      3. remove-double-neg 32.8%

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

      4. fma-neg 32.8%

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

      5. sub-neg 32.8%

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

      6. metadata-eval 32.8%

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

      7. remove-double-neg 32.8%

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

      8. fma-def 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around -inf 82.1%

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

      1. mul-1-neg 82.1%

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

      2. distribute-rgt-neg-in 82.1%

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

      3. +-commutative 82.1%

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

      4. sub-neg 82.1%

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

      5. metadata-eval 82.1%

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

      6. +-commutative 82.1%

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

   6. Simplified 82.1%

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

   7. Taylor expanded in x around inf 82.1%

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

      1. associate--r+ 82.1%

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

      2. sub-neg 82.1%

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

      3. metadata-eval 82.1%

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

      4. associate-*r/ 82.1%

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

      5. metadata-eval 82.1%

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

      6. unpow2 82.1%

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

   9. Simplified 82.1%

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

   if -1.49999999999999996e-223 < t < 2.1e-305

   1. Initial program 7.2%

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

      1. associate-*l/ 7.2%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 7.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 7.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 7.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 7.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 7.1%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

      3. metadata-eval 7.1%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 7.1%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      5. associate-/l* 7.2%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg 7.2%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      7. metadata-eval 7.2%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      8. +-commutative 7.2%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. Simplified 7.2%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

   8. Taylor expanded in x around inf 71.8%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{\left(\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}}} \cdot t \]
   9. Step-by-step derivation

      1. distribute-lft-in 71.8%

         \[\leadsto \sqrt{\frac{2}{\left(\frac{{\ell}^{2}}{x} + \color{blue}{2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right)}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}}} \cdot t \]

      2. associate--l+ 71.8%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{{\ell}^{2}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}}} \cdot t \]

      3. unpow2 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(2 \cdot \left(\frac{{t}^{2}}{x} + {t}^{2}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      4. +-commutative 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \color{blue}{\left({t}^{2} + \frac{{t}^{2}}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(\color{blue}{t \cdot t} + \frac{{t}^{2}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      6. unpow2 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \left(t \cdot t + \frac{\color{blue}{t \cdot t}}{x}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      7. fma-def 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \color{blue}{\mathsf{fma}\left(t, t, \frac{t \cdot t}{x}\right)} - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      8. associate-/l* 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{\frac{x}{t}}}\right) - -1 \cdot \frac{{\ell}^{2} + 2 \cdot {t}^{2}}{x}\right)}} \cdot t \]

      9. associate-*r/ 71.8%

         \[\leadsto \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \color{blue}{\frac{-1 \cdot \left({\ell}^{2} + 2 \cdot {t}^{2}\right)}{x}}\right)}} \cdot t \]

   10. Simplified 71.8%

       \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]
   11. Step-by-step derivation

       1. *-un-lft-identity 71.8%

          \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\right)} \cdot t \]

       2. associate-/l* 71.8%

          \[\leadsto \left(1 \cdot \sqrt{\frac{2}{\color{blue}{\frac{\ell}{\frac{x}{\ell}}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{\frac{x}{t}}\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\right) \cdot t \]

       3. associate-/r/ 71.8%

          \[\leadsto \left(1 \cdot \sqrt{\frac{2}{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \color{blue}{\frac{t}{x} \cdot t}\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\right) \cdot t \]

   12. Applied egg-rr 71.8%

       \[\leadsto \color{blue}{\left(1 \cdot \sqrt{\frac{2}{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}\right)} \cdot t \]
   13. Step-by-step derivation

       1. *-lft-identity 71.8%

          \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

       2. associate-/r/ 71.8%

          \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{\ell}{x} \cdot \ell} + \left(2 \cdot \mathsf{fma}\left(t, t, \frac{t}{x} \cdot t\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

       3. fma-def 71.8%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot t + \frac{t}{x} \cdot t\right)} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

       4. +-commutative 71.8%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(\frac{t}{x} \cdot t + t \cdot t\right)} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

       5. distribute-rgt-out 71.8%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \color{blue}{\left(t \cdot \left(\frac{t}{x} + t\right)\right)} - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}} \cdot t \]

   14. Simplified 71.8%

       \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(2 \cdot \left(t \cdot \left(\frac{t}{x} + t\right)\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\right)}}} \cdot t \]

   15. Taylor expanded in t around inf 53.1%

       \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \color{blue}{\left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right) \cdot {t}^{2}}}} \cdot t \]
   16. Step-by-step derivation

       1. *-commutative 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \color{blue}{{t}^{2} \cdot \left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)}}} \cdot t \]

       2. unpow2 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \color{blue}{\left(t \cdot t\right)} \cdot \left(2 \cdot \left(1 + \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)}} \cdot t \]

       3. distribute-lft-in 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(t \cdot t\right) \cdot \left(\color{blue}{\left(2 \cdot 1 + 2 \cdot \frac{1}{x}\right)} + 2 \cdot \frac{1}{x}\right)}} \cdot t \]

       4. metadata-eval 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(t \cdot t\right) \cdot \left(\left(\color{blue}{2} + 2 \cdot \frac{1}{x}\right) + 2 \cdot \frac{1}{x}\right)}} \cdot t \]

       5. associate-*r/ 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(t \cdot t\right) \cdot \left(\left(2 + \color{blue}{\frac{2 \cdot 1}{x}}\right) + 2 \cdot \frac{1}{x}\right)}} \cdot t \]

       6. metadata-eval 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(t \cdot t\right) \cdot \left(\left(2 + \frac{\color{blue}{2}}{x}\right) + 2 \cdot \frac{1}{x}\right)}} \cdot t \]

       7. associate-*r/ 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(t \cdot t\right) \cdot \left(\left(2 + \frac{2}{x}\right) + \color{blue}{\frac{2 \cdot 1}{x}}\right)}} \cdot t \]

       8. metadata-eval 53.1%

          \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \left(t \cdot t\right) \cdot \left(\left(2 + \frac{2}{x}\right) + \frac{\color{blue}{2}}{x}\right)}} \cdot t \]

   17. Simplified 53.1%

       \[\leadsto \sqrt{\frac{2}{\frac{\ell}{x} \cdot \ell + \color{blue}{\left(t \cdot t\right) \cdot \left(\left(2 + \frac{2}{x}\right) + \frac{2}{x}\right)}}} \cdot t \]

   if 2.1e-305 < t

   1. Initial program 24.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 71.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 71.4%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate--l+ 71.4%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]

      2. associate-*r/ 71.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]

      3. metadata-eval 71.4%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]

      4. unpow2 71.4%

         \[\leadsto 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-223}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-305}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\ell \cdot \frac{\ell}{x} + \left(t \cdot t\right) \cdot \left(\frac{2}{x} + \left(2 + \frac{2}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -8 \cdot 10^{-224}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - t_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{1 + x}{\frac{0.5}{t} \cdot \frac{x + -1}{t}} - \ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -8e-224)
     (- (+ (/ 1.0 x) -1.0) t_1)
     (if (<= t -3.8e-308)
       (*
        t
        (sqrt
         (/ 2.0 (- (/ (+ 1.0 x) (* (/ 0.5 t) (/ (+ x -1.0) t))) (* l l)))))
       (+ 1.0 (+ t_1 (/ -1.0 x)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -8e-224) {
		tmp = ((1.0 / x) + -1.0) - t_1;
	} else if (t <= -3.8e-308) {
		tmp = t * sqrt((2.0 / (((1.0 + x) / ((0.5 / t) * ((x + -1.0) / t))) - (l * l))));
	} else {
		tmp = 1.0 + (t_1 + (-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) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-8d-224)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) - t_1
    else if (t <= (-3.8d-308)) then
        tmp = t * sqrt((2.0d0 / (((1.0d0 + x) / ((0.5d0 / t) * ((x + (-1.0d0)) / t))) - (l * l))))
    else
        tmp = 1.0d0 + (t_1 + ((-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 t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -8e-224) {
		tmp = ((1.0 / x) + -1.0) - t_1;
	} else if (t <= -3.8e-308) {
		tmp = t * Math.sqrt((2.0 / (((1.0 + x) / ((0.5 / t) * ((x + -1.0) / t))) - (l * l))));
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -8e-224:
		tmp = ((1.0 / x) + -1.0) - t_1
	elif t <= -3.8e-308:
		tmp = t * math.sqrt((2.0 / (((1.0 + x) / ((0.5 / t) * ((x + -1.0) / t))) - (l * l))))
	else:
		tmp = 1.0 + (t_1 + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -8e-224)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - t_1);
	elseif (t <= -3.8e-308)
		tmp = Float64(t * sqrt(Float64(2.0 / Float64(Float64(Float64(1.0 + x) / Float64(Float64(0.5 / t) * Float64(Float64(x + -1.0) / t))) - Float64(l * l)))));
	else
		tmp = Float64(1.0 + Float64(t_1 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -8e-224)
		tmp = ((1.0 / x) + -1.0) - t_1;
	elseif (t <= -3.8e-308)
		tmp = t * sqrt((2.0 / (((1.0 + x) / ((0.5 / t) * ((x + -1.0) / t))) - (l * l))));
	else
		tmp = 1.0 + (t_1 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8e-224], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[t, -3.8e-308], N[(t * N[Sqrt[N[(2.0 / N[(N[(N[(1.0 + x), $MachinePrecision] / N[(N[(0.5 / t), $MachinePrecision] * N[(N[(x + -1.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.0000000000000002e-224

   1. Initial program 32.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 32.8%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 32.8%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 32.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 82.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 82.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 82.1%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 82.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 82.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 82.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 82.1%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 82.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 82.1%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 82.1%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

      2. sub-neg 82.1%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

      3. metadata-eval 82.1%

         \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 82.1%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 82.1%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      6. unpow2 82.1%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

   9. Simplified 82.1%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

   if -8.0000000000000002e-224 < t < -3.79999999999999975e-308

   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-*l/ 1.2%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 1.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 1.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 1.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 1.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 1.0%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

      3. metadata-eval 1.0%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 1.0%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      5. associate-/l* 0.6%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg 0.6%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      7. metadata-eval 0.6%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      8. +-commutative 0.6%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. Simplified 0.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

   8. Taylor expanded in t around inf 39.5%

      \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\color{blue}{0.5 \cdot \frac{x - 1}{{t}^{2}}}} - \ell \cdot \ell}} \cdot t \]
   9. Step-by-step derivation

      1. associate-*r/ 39.5%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\color{blue}{\frac{0.5 \cdot \left(x - 1\right)}{{t}^{2}}}} - \ell \cdot \ell}} \cdot t \]

      2. unpow2 39.5%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{0.5 \cdot \left(x - 1\right)}{\color{blue}{t \cdot t}}} - \ell \cdot \ell}} \cdot t \]

      3. times-frac 39.5%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\color{blue}{\frac{0.5}{t} \cdot \frac{x - 1}{t}}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 39.5%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{0.5}{t} \cdot \frac{\color{blue}{x + \left(-1\right)}}{t}} - \ell \cdot \ell}} \cdot t \]

      5. metadata-eval 39.5%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{0.5}{t} \cdot \frac{x + \color{blue}{-1}}{t}} - \ell \cdot \ell}} \cdot t \]

   10. Simplified 39.5%

       \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\color{blue}{\frac{0.5}{t} \cdot \frac{x + -1}{t}}} - \ell \cdot \ell}} \cdot t \]

   if -3.79999999999999975e-308 < t

   1. Initial program 24.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.5%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 69.8%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 70.1%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate--l+ 70.1%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]

      2. associate-*r/ 70.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]

      3. metadata-eval 70.1%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]

      4. unpow2 70.1%

         \[\leadsto 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]

   7. Simplified 70.1%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-224}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{1 + x}{\frac{0.5}{t} \cdot \frac{x + -1}{t}} - \ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10: 76.4% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ (/ 1.0 x) -1.0) (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-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 <= (-5d-310)) then
        tmp = (1.0d0 / x) + (-1.0d0)
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-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 <= -5e-310) {
		tmp = (1.0 / x) + -1.0;
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = (1.0 / x) + -1.0
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(1.0 / x) + -1.0);
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = (1.0 / x) + -1.0;
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-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, -5e-310], N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 29.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 29.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 73.4%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if -4.999999999999985e-310 < t

   1. Initial program 24.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 70.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 70.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate--l+ 70.6%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]

      2. associate-*r/ 70.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]

      3. metadata-eval 70.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]

      4. unpow2 70.6%

         \[\leadsto 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 11: 76.5% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -5e-310) (- (+ (/ 1.0 x) -1.0) t_1) (+ 1.0 (+ t_1 (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -5e-310) {
		tmp = ((1.0 / x) + -1.0) - t_1;
	} else {
		tmp = 1.0 + (t_1 + (-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) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-5d-310)) then
        tmp = ((1.0d0 / x) + (-1.0d0)) - t_1
    else
        tmp = 1.0d0 + (t_1 + ((-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 t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -5e-310) {
		tmp = ((1.0 / x) + -1.0) - t_1;
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -5e-310:
		tmp = ((1.0 / x) + -1.0) - t_1
	else:
		tmp = 1.0 + (t_1 + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(Float64(Float64(1.0 / x) + -1.0) - t_1);
	else
		tmp = Float64(1.0 + Float64(t_1 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = ((1.0 / x) + -1.0) - t_1;
	else
		tmp = 1.0 + (t_1 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e-310], N[(N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision] - t$95$1), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 29.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 29.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 73.6%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 73.6%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

      2. sub-neg 73.6%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

      3. metadata-eval 73.6%

         \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 73.6%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 73.6%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      6. unpow2 73.6%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

   9. Simplified 73.6%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

   if -4.999999999999985e-310 < t

   1. Initial program 24.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 70.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 70.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{x}} \]
   6. Step-by-step derivation

      1. associate--l+ 70.6%

         \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \frac{1}{x}\right)} \]

      2. associate-*r/ 70.6%

         \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} - \frac{1}{x}\right) \]

      3. metadata-eval 70.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} - \frac{1}{x}\right) \]

      4. unpow2 70.6%

         \[\leadsto 1 + \left(\frac{0.5}{\color{blue}{x \cdot x}} - \frac{1}{x}\right) \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{1 + \left(\frac{0.5}{x \cdot x} - \frac{1}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 12: 75.9% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) -1.0 (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], -1.0, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 29.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 29.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 72.9%

      \[\leadsto \color{blue}{-1} \]

   if -4.999999999999985e-310 < t

   1. Initial program 24.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 70.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 70.5%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13: 76.3% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -5e-310) (+ (/ 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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 29.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 29.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 73.4%

      \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

   if -4.999999999999985e-310 < t

   1. Initial program 24.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 70.3%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]

   5. Taylor expanded in x around inf 70.5%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{x} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 14: 75.6% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -5e-310) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 29.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-/l* 29.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

      2. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

      3. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

      4. fma-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

      5. sub-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      6. metadata-eval 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

      7. remove-double-neg 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

      8. fma-def 29.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   3. Simplified 29.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. mul-1-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. distribute-rgt-neg-in 73.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

      3. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

      5. metadata-eval 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

      6. +-commutative 73.6%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

   6. Simplified 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

   7. Taylor expanded in x around inf 72.9%

      \[\leadsto \color{blue}{-1} \]

   if -4.999999999999985e-310 < t

   1. Initial program 24.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.7%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 68.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   5. Step-by-step derivation

      1. sqrt-unprod 69.7%

         \[\leadsto \color{blue}{\sqrt{2 \cdot 0.5}} \]

      2. metadata-eval 69.7%

         \[\leadsto \sqrt{\color{blue}{1}} \]

      3. metadata-eval 69.7%

         \[\leadsto \color{blue}{1} \]

   6. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 38.1% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 27.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-/l* 27.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

   2. fma-neg 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \ell \cdot \ell + 2 \cdot \left(t \cdot t\right), -\ell \cdot \ell\right)}}}{t}} \]

   3. remove-double-neg 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \color{blue}{-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)}, -\ell \cdot \ell\right)}}{t}} \]

   4. fma-neg 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{x + 1}{x - 1} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}}{t}} \]

   5. sub-neg 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

   6. metadata-eval 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + \color{blue}{-1}} \cdot \left(-\left(-\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)\right)\right) - \ell \cdot \ell}}{t}} \]

   7. remove-double-neg 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

   8. fma-def 27.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \color{blue}{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)} - \ell \cdot \ell}}{t}} \]

3. Simplified 27.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}{t}}} \]

4. Taylor expanded in t around -inf 37.7%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
5. Step-by-step derivation

   1. mul-1-neg 37.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. distribute-rgt-neg-in 37.7%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{1 + x}{x - 1}}\right)}} \]

   3. +-commutative 37.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{\color{blue}{x + 1}}{x - 1}}\right)} \]

   4. sub-neg 37.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}\right)} \]

   5. metadata-eval 37.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{x + \color{blue}{-1}}}\right)} \]

   6. +-commutative 37.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}\right)} \]

6. Simplified 37.7%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}} \]

7. Taylor expanded in x around inf 37.3%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 37.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023228 
(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)))))