Toniolo and Linder, Equation (7)

Percentage Accurate: 34.3% → 84.3%

Time: 26.9s

Alternatives: 13

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: 34.3% 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: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + t_1}}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-206}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t \cdot t\right)}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \left(t_1 + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (fma 2.0 (* t t) (* l l)) x)))
   (if (<= t -4.8e+60)
     -1.0
     (if (<= t -1e-179)
       (*
        t
        (sqrt
         (/
          2.0
          (+ (fma 2.0 (/ (* t t) x) (fma 2.0 (* t t) (/ (* l l) x))) t_1))))
       (if (<= t -1.35e-206)
         -1.0
         (if (<= t 1.3e-216)
           (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
           (if (<= t 2.55e-147)
             1.0
             (if (<= t 3.2e+73)
               (/
                (sqrt (* 2.0 (* t t)))
                (sqrt (+ (* 2.0 (* t (+ t (/ t x)))) (+ t_1 (* l (/ l x))))))
               (*
                t
                (/
                 (sqrt 2.0)
                 (* (* t (sqrt 2.0)) (sqrt (/ (+ x 1.0) (+ -1.0 x))))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = fma(2.0, (t * t), (l * l)) / x;
	double tmp;
	if (t <= -4.8e+60) {
		tmp = -1.0;
	} else if (t <= -1e-179) {
		tmp = t * sqrt((2.0 / (fma(2.0, ((t * t) / x), fma(2.0, (t * t), ((l * l) / x))) + t_1)));
	} else if (t <= -1.35e-206) {
		tmp = -1.0;
	} else if (t <= 1.3e-216) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 3.2e+73) {
		tmp = sqrt((2.0 * (t * t))) / sqrt(((2.0 * (t * (t + (t / x)))) + (t_1 + (l * (l / x)))));
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 1.0) / (-1.0 + x)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x)
	tmp = 0.0
	if (t <= -4.8e+60)
		tmp = -1.0;
	elseif (t <= -1e-179)
		tmp = Float64(t * sqrt(Float64(2.0 / Float64(fma(2.0, Float64(Float64(t * t) / x), fma(2.0, Float64(t * t), Float64(Float64(l * l) / x))) + t_1))));
	elseif (t <= -1.35e-206)
		tmp = -1.0;
	elseif (t <= 1.3e-216)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 3.2e+73)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t * t))) / sqrt(Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(t_1 + Float64(l * Float64(l / x))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -4.8e+60], -1.0, If[LessEqual[t, -1e-179], N[(t * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.35e-206], -1.0, If[LessEqual[t, 1.3e-216], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 3.2e+73], N[(N[Sqrt[N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.8e60 or -1e-179 < t < -1.35e-206

   1. Initial program 27.3%

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

      1. associate-*l/ 27.3%

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

   3. Simplified 27.3%

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

      1. expm1-log1p-u 27.3%

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

      2. expm1-udef 2.4%

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

   5. Applied egg-rr 2.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   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* 27.4%

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

      6. sub-neg 27.4%

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

      7. metadata-eval 27.4%

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

      8. +-commutative 27.4%

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

   7. Simplified 27.4%

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

   8. Taylor expanded in x around inf 35.9%

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

      1. unpow2 35.9%

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

   10. Simplified 35.9%

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

   11. Taylor expanded in t around -inf 95.4%

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

   if -4.8e60 < t < -1e-179

   1. Initial program 48.8%

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

      1. associate-*l/ 48.8%

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

   3. Simplified 48.8%

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

      1. expm1-log1p-u 47.0%

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

      2. expm1-udef 30.6%

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

   5. Applied egg-rr 30.7%

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

      1. expm1-def 43.1%

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

      2. expm1-log1p 45.0%

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

      3. metadata-eval 45.0%

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

      4. sub-neg 45.0%

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

      5. associate-/l* 36.8%

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

      6. sub-neg 36.8%

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

      7. metadata-eval 36.8%

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

      8. +-commutative 36.8%

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

   7. Simplified 36.8%

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

   8. Taylor expanded in x around inf 87.3%

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

      1. sub-neg 87.3%

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

      2. fma-def 87.3%

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

      3. unpow2 87.3%

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

      4. fma-def 87.3%

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

      5. unpow2 87.3%

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

      6. unpow2 87.3%

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

      7. mul-1-neg 87.3%

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

      8. unpow2 87.3%

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

      9. unpow2 87.3%

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

      10. fma-udef 87.3%

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

      11. remove-double-neg 87.3%

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

   10. Simplified 87.3%

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

   if -1.35e-206 < t < 1.2999999999999999e-216

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. sub-neg 55.4%

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

      2. fma-def 55.4%

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

      3. unpow2 55.4%

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

      4. fma-def 55.4%

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

      5. unpow2 55.4%

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

      6. unpow2 55.4%

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

      7. mul-1-neg 55.4%

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

      8. remove-double-neg 55.4%

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

      9. fma-def 55.4%

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

      10. unpow2 55.4%

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

      11. unpow2 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in t around 0 47.6%

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

      1. associate-*l* 47.6%

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

   9. Simplified 47.6%

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

   if 1.2999999999999999e-216 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 3.19999999999999982e73

   1. Initial program 58.1%

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

      1. add-sqr-sqrt 58.0%

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

      2. sqrt-prod 58.1%

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

      3. sqrt-prod 58.5%

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

      4. pow1/2 58.5%

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

   3. Applied egg-rr 58.5%

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

      1. unpow1/2 58.5%

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

   5. Simplified 58.5%

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

   6. Taylor expanded in x around inf 87.5%

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

      1. sub-neg 87.5%

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

      2. associate-+r+ 87.5%

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

      3. mul-1-neg 87.5%

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

      4. unpow2 87.5%

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

      5. unpow2 87.5%

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

      6. fma-udef 87.5%

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

      7. remove-double-neg 87.5%

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

      8. associate-+l+ 87.5%

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

   8. Simplified 87.5%

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

   if 3.19999999999999982e73 < t

   1. Initial program 30.3%

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

      1. associate-*l/ 30.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in t around inf 91.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-206}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t \cdot t\right)}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-207}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t \cdot t\right)}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -2e+58)
   -1.0
   (if (<= t -3.9e-175)
     (*
      t
      (/
       (sqrt 2.0)
       (sqrt (* 2.0 (+ (/ (* t (* t 2.0)) x) (+ (* t t) (/ (* l l) x)))))))
     (if (<= t -9.2e-207)
       -1.0
       (if (<= t 6.8e-217)
         (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
         (if (<= t 2.55e-147)
           1.0
           (if (<= t 2.8e+73)
             (/
              (sqrt (* 2.0 (* t t)))
              (sqrt
               (+
                (* 2.0 (* t (+ t (/ t x))))
                (+ (/ (fma 2.0 (* t t) (* l l)) x) (* l (/ l x))))))
             (*
              t
              (/
               (sqrt 2.0)
               (* (* t (sqrt 2.0)) (sqrt (/ (+ x 1.0) (+ -1.0 x)))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e+58) {
		tmp = -1.0;
	} else if (t <= -3.9e-175) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else if (t <= -9.2e-207) {
		tmp = -1.0;
	} else if (t <= 6.8e-217) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 2.8e+73) {
		tmp = sqrt((2.0 * (t * t))) / sqrt(((2.0 * (t * (t + (t / x)))) + ((fma(2.0, (t * t), (l * l)) / x) + (l * (l / x)))));
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 1.0) / (-1.0 + x)))));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2e+58)
		tmp = -1.0;
	elseif (t <= -3.9e-175)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(Float64(t * Float64(t * 2.0)) / x) + Float64(Float64(t * t) + Float64(Float64(l * l) / x)))))));
	elseif (t <= -9.2e-207)
		tmp = -1.0;
	elseif (t <= 6.8e-217)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 2.8e+73)
		tmp = Float64(sqrt(Float64(2.0 * Float64(t * t))) / sqrt(Float64(Float64(2.0 * Float64(t * Float64(t + Float64(t / x)))) + Float64(Float64(fma(2.0, Float64(t * t), Float64(l * l)) / x) + Float64(l * Float64(l / x))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))))));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -2e+58], -1.0, If[LessEqual[t, -3.9e-175], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -9.2e-207], -1.0, If[LessEqual[t, 6.8e-217], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 2.8e+73], N[(N[Sqrt[N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(2.0 * N[(t * N[(t + N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.99999999999999989e58 or -3.89999999999999998e-175 < t < -9.2000000000000002e-207

   1. Initial program 28.4%

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

      1. associate-*l/ 28.4%

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

   3. Simplified 28.4%

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

      1. expm1-log1p-u 28.4%

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

      2. expm1-udef 2.4%

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

   5. Applied egg-rr 2.4%

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

      1. expm1-def 7.0%

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

      2. expm1-log1p 7.0%

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

      3. metadata-eval 7.0%

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

      4. sub-neg 7.0%

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

      5. associate-/l* 28.5%

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

      6. sub-neg 28.5%

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

      7. metadata-eval 28.5%

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

      8. +-commutative 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in x around inf 37.0%

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

      1. unpow2 37.0%

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

   10. Simplified 37.0%

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

   11. Taylor expanded in t around -inf 95.5%

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

   if -1.99999999999999989e58 < t < -3.89999999999999998e-175

   1. Initial program 47.8%

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

      1. associate-*l/ 47.7%

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

   3. Simplified 47.7%

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

   4. Taylor expanded in l around 0 69.3%

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

      1. fma-def 69.3%

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

      2. +-commutative 69.3%

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

      3. associate-*r/ 73.5%

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

      4. sub-neg 73.5%

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

      5. metadata-eval 73.5%

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

      6. unpow2 73.5%

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

      7. +-commutative 73.5%

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

      8. unpow2 73.5%

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

      9. associate--l+ 76.2%

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

      10. sub-neg 76.2%

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

      11. metadata-eval 76.2%

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

      12. +-commutative 76.2%

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

      13. sub-neg 76.2%

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

      14. metadata-eval 76.2%

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

      15. +-commutative 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in x around inf 86.9%

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

      1. distribute-lft-out 86.9%

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

      2. distribute-lft-out 86.9%

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

      3. cancel-sign-sub-inv 86.9%

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

      4. metadata-eval 86.9%

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

      5. distribute-rgt1-in 86.9%

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

      6. metadata-eval 86.9%

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

      7. *-commutative 86.9%

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

      8. unpow2 86.9%

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

      9. associate-*l* 86.9%

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

      10. unpow2 86.9%

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

      11. unpow2 86.9%

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

   9. Simplified 86.9%

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

   if -9.2000000000000002e-207 < t < 6.80000000000000032e-217

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. sub-neg 55.4%

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

      2. fma-def 55.4%

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

      3. unpow2 55.4%

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

      4. fma-def 55.4%

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

      5. unpow2 55.4%

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

      6. unpow2 55.4%

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

      7. mul-1-neg 55.4%

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

      8. remove-double-neg 55.4%

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

      9. fma-def 55.4%

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

      10. unpow2 55.4%

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

      11. unpow2 55.4%

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

   6. Simplified 55.4%

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

   7. Taylor expanded in t around 0 47.6%

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

      1. associate-*l* 47.6%

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

   9. Simplified 47.6%

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

   if 6.80000000000000032e-217 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 2.80000000000000008e73

   1. Initial program 58.1%

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

      1. add-sqr-sqrt 58.0%

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

      2. sqrt-prod 58.1%

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

      3. sqrt-prod 58.5%

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

      4. pow1/2 58.5%

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

   3. Applied egg-rr 58.5%

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

      1. unpow1/2 58.5%

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

   5. Simplified 58.5%

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

   6. Taylor expanded in x around inf 87.5%

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

      1. sub-neg 87.5%

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

      2. associate-+r+ 87.5%

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

      3. mul-1-neg 87.5%

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

      4. unpow2 87.5%

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

      5. unpow2 87.5%

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

      6. fma-udef 87.5%

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

      7. remove-double-neg 87.5%

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

      8. associate-+l+ 87.5%

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

   8. Simplified 87.5%

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

   if 2.80000000000000008e73 < t

   1. Initial program 30.3%

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

      1. associate-*l/ 30.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in t around inf 91.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \cdot t \]
3. Recombined 6 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+58}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq -3.9 \cdot 10^{-175}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-207}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(t \cdot t\right)}}{\sqrt{2 \cdot \left(t \cdot \left(t + \frac{t}{x}\right)\right) + \left(\frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 3: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\ell \cdot \ell}{x}\\ t_2 := t \cdot \sqrt{2}\\ t_3 := \sqrt{\frac{x + 1}{-1 + x}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_2 \cdot \left(-t_3\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{t_1 + \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, t_1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_2 \cdot t_3}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (* l l) x))
        (t_2 (* t (sqrt 2.0)))
        (t_3 (sqrt (/ (+ x 1.0) (+ -1.0 x)))))
   (if (<= t -1.02e-150)
     (* t (/ (sqrt 2.0) (* t_2 (- t_3))))
     (if (<= t 3.5e-217)
       (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
       (if (<= t 2.55e-147)
         1.0
         (if (<= t 7e+73)
           (*
            t
            (sqrt
             (/ 2.0 (+ t_1 (fma 2.0 (/ (* t t) x) (fma 2.0 (* t t) t_1))))))
           (* t (/ (sqrt 2.0) (* t_2 t_3)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (l * l) / x;
	double t_2 = t * sqrt(2.0);
	double t_3 = sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -1.02e-150) {
		tmp = t * (sqrt(2.0) / (t_2 * -t_3));
	} else if (t <= 3.5e-217) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 7e+73) {
		tmp = t * sqrt((2.0 / (t_1 + fma(2.0, ((t * t) / x), fma(2.0, (t * t), t_1)))));
	} else {
		tmp = t * (sqrt(2.0) / (t_2 * t_3));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(l * l) / x)
	t_2 = Float64(t * sqrt(2.0))
	t_3 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	tmp = 0.0
	if (t <= -1.02e-150)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_2 * Float64(-t_3))));
	elseif (t <= 3.5e-217)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 7e+73)
		tmp = Float64(t * sqrt(Float64(2.0 / Float64(t_1 + fma(2.0, Float64(Float64(t * t) / x), fma(2.0, Float64(t * t), t_1))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_2 * t_3)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.02e-150], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 * (-t$95$3)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e-217], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 7e+73], N[(t * N[Sqrt[N[(2.0 / N[(t$95$1 + N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.0199999999999999e-150

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around -inf 89.0%

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

      1. mul-1-neg 89.0%

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

      2. *-commutative 89.0%

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

      3. distribute-rgt-neg-in 89.0%

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

      4. +-commutative 89.0%

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

      5. sub-neg 89.0%

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

      6. metadata-eval 89.0%

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

      7. +-commutative 89.0%

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

      8. distribute-rgt-neg-in 89.0%

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

   6. Simplified 89.0%

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

   if -1.0199999999999999e-150 < t < 3.5e-217

   1. Initial program 2.1%

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

      1. associate-*l/ 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in x around inf 56.0%

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

      1. sub-neg 56.0%

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

      2. fma-def 56.0%

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

      3. unpow2 56.0%

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

      4. fma-def 56.0%

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

      5. unpow2 56.0%

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

      6. unpow2 56.0%

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

      7. mul-1-neg 56.0%

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

      8. remove-double-neg 56.0%

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

      9. fma-def 56.0%

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

      10. unpow2 56.0%

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

      11. unpow2 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in t around 0 41.0%

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

      1. associate-*l* 41.0%

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

   9. Simplified 41.0%

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

   if 3.5e-217 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 7.00000000000000004e73

   1. Initial program 58.1%

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

      1. associate-*l/ 58.3%

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

   3. Simplified 58.3%

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

      1. expm1-log1p-u 56.7%

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

      2. expm1-udef 25.1%

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

   5. Applied egg-rr 25.1%

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

      1. expm1-def 39.3%

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

      2. expm1-log1p 41.0%

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

      3. metadata-eval 41.0%

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

      4. sub-neg 41.0%

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

      5. associate-/l* 47.4%

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

      6. sub-neg 47.4%

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

      7. metadata-eval 47.4%

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

      8. +-commutative 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in x around inf 87.3%

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

      1. sub-neg 87.3%

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

      2. fma-def 87.3%

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

      3. unpow2 87.3%

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

      4. fma-def 87.3%

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

      5. unpow2 87.3%

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

      6. unpow2 87.3%

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

      7. mul-1-neg 87.3%

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

      8. unpow2 87.3%

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

      9. unpow2 87.3%

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

      10. fma-udef 87.3%

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

      11. remove-double-neg 87.3%

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

   10. Simplified 87.3%

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

   11. Taylor expanded in t around 0 87.3%

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

       1. unpow2 87.3%

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

   13. Simplified 87.3%

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

   if 7.00000000000000004e73 < t

   1. Initial program 30.3%

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

      1. associate-*l/ 30.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in t around inf 91.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \sqrt{\frac{2}{\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 4: 82.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.35e-150)
   (* t (/ (sqrt 2.0) (* t (- (sqrt (+ 2.0 (/ 4.0 x)))))))
   (if (<= t 1.2e-217)
     (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
     (if (<= t 2.55e-147)
       1.0
       (if (<= t 1.1e+43)
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (* 2.0 (+ (/ (* t (* t 2.0)) x) (+ (* t t) (/ (* l l) x)))))))
         (*
          t
          (/
           (sqrt 2.0)
           (* (* t (sqrt 2.0)) (sqrt (/ (+ x 1.0) (+ -1.0 x)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.35e-150) {
		tmp = t * (sqrt(2.0) / (t * -sqrt((2.0 + (4.0 / x)))));
	} else if (t <= 1.2e-217) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 1.1e+43) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 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 <= (-1.35d-150)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt((2.0d0 + (4.0d0 / x)))))
    else if (t <= 1.2d-217) then
        tmp = t * (sqrt(2.0d0) / (l * (sqrt(2.0d0) * sqrt((1.0d0 / x)))))
    else if (t <= 2.55d-147) then
        tmp = 1.0d0
    else if (t <= 1.1d+43) then
        tmp = t * (sqrt(2.0d0) / sqrt((2.0d0 * (((t * (t * 2.0d0)) / x) + ((t * t) + ((l * l) / x))))))
    else
        tmp = t * (sqrt(2.0d0) / ((t * sqrt(2.0d0)) * sqrt(((x + 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 <= -1.35e-150) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt((2.0 + (4.0 / x)))));
	} else if (t <= 1.2e-217) {
		tmp = t * (Math.sqrt(2.0) / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 1.1e+43) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = t * (Math.sqrt(2.0) / ((t * Math.sqrt(2.0)) * Math.sqrt(((x + 1.0) / (-1.0 + x)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.35e-150:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt((2.0 + (4.0 / x)))))
	elif t <= 1.2e-217:
		tmp = t * (math.sqrt(2.0) / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t <= 2.55e-147:
		tmp = 1.0
	elif t <= 1.1e+43:
		tmp = t * (math.sqrt(2.0) / math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))))
	else:
		tmp = t * (math.sqrt(2.0) / ((t * math.sqrt(2.0)) * math.sqrt(((x + 1.0) / (-1.0 + x)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.35e-150)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(2.0 + Float64(4.0 / x)))))));
	elseif (t <= 1.2e-217)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 1.1e+43)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(Float64(t * Float64(t * 2.0)) / x) + Float64(Float64(t * t) + Float64(Float64(l * l) / x)))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * sqrt(Float64(Float64(x + 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 <= -1.35e-150)
		tmp = t * (sqrt(2.0) / (t * -sqrt((2.0 + (4.0 / x)))));
	elseif (t <= 1.2e-217)
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 1.1e+43)
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	else
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * sqrt(((x + 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, -1.35e-150], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.2e-217], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 1.1e+43], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.3500000000000001e-150

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. sub-neg 56.8%

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

      2. fma-def 56.8%

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

      3. unpow2 56.8%

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

      4. fma-def 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. mul-1-neg 56.8%

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

      8. remove-double-neg 56.8%

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

      9. fma-def 56.8%

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

      10. unpow2 56.8%

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

      11. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in t around -inf 88.8%

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

      1. mul-1-neg 88.8%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]

      2. distribute-rgt-neg-in 88.8%

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

      3. associate-*r/ 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -1.3500000000000001e-150 < t < 1.1999999999999999e-217

   1. Initial program 2.1%

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

      1. associate-*l/ 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in x around inf 56.0%

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

      1. sub-neg 56.0%

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

      2. fma-def 56.0%

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

      3. unpow2 56.0%

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

      4. fma-def 56.0%

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

      5. unpow2 56.0%

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

      6. unpow2 56.0%

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

      7. mul-1-neg 56.0%

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

      8. remove-double-neg 56.0%

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

      9. fma-def 56.0%

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

      10. unpow2 56.0%

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

      11. unpow2 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in t around 0 41.0%

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

      1. associate-*l* 41.0%

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

   9. Simplified 41.0%

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

   if 1.1999999999999999e-217 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 1.1e43

   1. Initial program 50.5%

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

      1. associate-*l/ 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in l around 0 53.5%

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

      1. fma-def 53.5%

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

      2. +-commutative 53.5%

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

      3. associate-*r/ 61.6%

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

      4. sub-neg 61.6%

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

      5. metadata-eval 61.6%

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

      6. unpow2 61.6%

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

      7. +-commutative 61.6%

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

      8. unpow2 61.6%

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

      9. associate--l+ 68.1%

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

      10. sub-neg 68.1%

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

      11. metadata-eval 68.1%

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

      12. +-commutative 68.1%

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

      13. sub-neg 68.1%

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

      14. metadata-eval 68.1%

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

      15. +-commutative 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. distribute-lft-out 84.0%

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

      2. distribute-lft-out 84.0%

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

      3. cancel-sign-sub-inv 84.0%

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

      4. metadata-eval 84.0%

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

      5. distribute-rgt1-in 84.0%

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

      6. metadata-eval 84.0%

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

      7. *-commutative 84.0%

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

      8. unpow2 84.0%

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

      9. associate-*l* 84.0%

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

      10. unpow2 84.0%

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

      11. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if 1.1e43 < t

   1. Initial program 37.7%

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

      1. associate-*l/ 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around inf 92.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+43}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 5: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + 1}{-1 + x}}\\ \mathbf{if}\;t \leq -6 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))) (t_2 (sqrt (/ (+ x 1.0) (+ -1.0 x)))))
   (if (<= t -6e-151)
     (* t (/ (sqrt 2.0) (* t_1 (- t_2))))
     (if (<= t 4.2e-217)
       (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
       (if (<= t 2.55e-147)
         1.0
         (if (<= t 1.35e+42)
           (*
            t
            (/
             (sqrt 2.0)
             (sqrt
              (* 2.0 (+ (/ (* t (* t 2.0)) x) (+ (* t t) (/ (* l l) x)))))))
           (* t (/ (sqrt 2.0) (* t_1 t_2)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -6e-151) {
		tmp = t * (sqrt(2.0) / (t_1 * -t_2));
	} else if (t <= 4.2e-217) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 1.35e+42) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = t * (sqrt(2.0) / (t_1 * t_2));
	}
	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) :: tmp
    t_1 = t * sqrt(2.0d0)
    t_2 = sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    if (t <= (-6d-151)) then
        tmp = t * (sqrt(2.0d0) / (t_1 * -t_2))
    else if (t <= 4.2d-217) then
        tmp = t * (sqrt(2.0d0) / (l * (sqrt(2.0d0) * sqrt((1.0d0 / x)))))
    else if (t <= 2.55d-147) then
        tmp = 1.0d0
    else if (t <= 1.35d+42) then
        tmp = t * (sqrt(2.0d0) / sqrt((2.0d0 * (((t * (t * 2.0d0)) / x) + ((t * t) + ((l * l) / x))))))
    else
        tmp = t * (sqrt(2.0d0) / (t_1 * t_2))
    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);
	double t_2 = Math.sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -6e-151) {
		tmp = t * (Math.sqrt(2.0) / (t_1 * -t_2));
	} else if (t <= 4.2e-217) {
		tmp = t * (Math.sqrt(2.0) / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 1.35e+42) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = t * (Math.sqrt(2.0) / (t_1 * t_2));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * math.sqrt(2.0)
	t_2 = math.sqrt(((x + 1.0) / (-1.0 + x)))
	tmp = 0
	if t <= -6e-151:
		tmp = t * (math.sqrt(2.0) / (t_1 * -t_2))
	elif t <= 4.2e-217:
		tmp = t * (math.sqrt(2.0) / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t <= 2.55e-147:
		tmp = 1.0
	elif t <= 1.35e+42:
		tmp = t * (math.sqrt(2.0) / math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))))
	else:
		tmp = t * (math.sqrt(2.0) / (t_1 * t_2))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	tmp = 0.0
	if (t <= -6e-151)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * Float64(-t_2))));
	elseif (t <= 4.2e-217)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 1.35e+42)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(Float64(t * Float64(t * 2.0)) / x) + Float64(Float64(t * t) + Float64(Float64(l * l) / x)))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * t_2)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * sqrt(2.0);
	t_2 = sqrt(((x + 1.0) / (-1.0 + x)));
	tmp = 0.0;
	if (t <= -6e-151)
		tmp = t * (sqrt(2.0) / (t_1 * -t_2));
	elseif (t <= 4.2e-217)
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 1.35e+42)
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	else
		tmp = t * (sqrt(2.0) / (t_1 * t_2));
	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[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -6e-151], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * (-t$95$2)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-217], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 1.35e+42], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6e-151

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in t around -inf 89.0%

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

      1. mul-1-neg 89.0%

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

      2. *-commutative 89.0%

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

      3. distribute-rgt-neg-in 89.0%

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

      4. +-commutative 89.0%

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

      5. sub-neg 89.0%

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

      6. metadata-eval 89.0%

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

      7. +-commutative 89.0%

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

      8. distribute-rgt-neg-in 89.0%

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

   6. Simplified 89.0%

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

   if -6e-151 < t < 4.2e-217

   1. Initial program 2.1%

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

      1. associate-*l/ 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in x around inf 56.0%

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

      1. sub-neg 56.0%

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

      2. fma-def 56.0%

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

      3. unpow2 56.0%

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

      4. fma-def 56.0%

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

      5. unpow2 56.0%

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

      6. unpow2 56.0%

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

      7. mul-1-neg 56.0%

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

      8. remove-double-neg 56.0%

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

      9. fma-def 56.0%

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

      10. unpow2 56.0%

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

      11. unpow2 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in t around 0 41.0%

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

      1. associate-*l* 41.0%

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

   9. Simplified 41.0%

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

   if 4.2e-217 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 1.35e42

   1. Initial program 50.5%

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

      1. associate-*l/ 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in l around 0 53.5%

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

      1. fma-def 53.5%

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

      2. +-commutative 53.5%

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

      3. associate-*r/ 61.6%

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

      4. sub-neg 61.6%

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

      5. metadata-eval 61.6%

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

      6. unpow2 61.6%

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

      7. +-commutative 61.6%

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

      8. unpow2 61.6%

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

      9. associate--l+ 68.1%

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

      10. sub-neg 68.1%

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

      11. metadata-eval 68.1%

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

      12. +-commutative 68.1%

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

      13. sub-neg 68.1%

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

      14. metadata-eval 68.1%

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

      15. +-commutative 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. distribute-lft-out 84.0%

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

      2. distribute-lft-out 84.0%

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

      3. cancel-sign-sub-inv 84.0%

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

      4. metadata-eval 84.0%

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

      5. distribute-rgt1-in 84.0%

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

      6. metadata-eval 84.0%

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

      7. *-commutative 84.0%

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

      8. unpow2 84.0%

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

      9. associate-*l* 84.0%

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

      10. unpow2 84.0%

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

      11. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if 1.35e42 < t

   1. Initial program 37.7%

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

      1. associate-*l/ 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in t around inf 92.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \left(-\sqrt{\frac{x + 1}{-1 + x}}\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 6: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -3.6e-151)
   (* t (/ (sqrt 2.0) (* t (- (sqrt (+ 2.0 (/ 4.0 x)))))))
   (if (<= t 1.4e-216)
     (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x)))))))
     (if (<= t 2.55e-147)
       1.0
       (if (<= t 8.8e+73)
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (* 2.0 (+ (/ (* t (* t 2.0)) x) (+ (* t t) (/ (* l l) x)))))))
         (* (* (sqrt 2.0) (sqrt 0.5)) (sqrt (/ (+ -1.0 x) (+ x 1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-151) {
		tmp = t * (sqrt(2.0) / (t * -sqrt((2.0 + (4.0 / x)))));
	} else if (t <= 1.4e-216) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 8.8e+73) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = (sqrt(2.0) * sqrt(0.5)) * sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.6d-151)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt((2.0d0 + (4.0d0 / x)))))
    else if (t <= 1.4d-216) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((1.0d0 / x) + (1.0d0 / ((-1.0d0) + x))))))
    else if (t <= 2.55d-147) then
        tmp = 1.0d0
    else if (t <= 8.8d+73) then
        tmp = t * (sqrt(2.0d0) / sqrt((2.0d0 * (((t * (t * 2.0d0)) / x) + ((t * t) + ((l * l) / x))))))
    else
        tmp = (sqrt(2.0d0) * sqrt(0.5d0)) * sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-151) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt((2.0 + (4.0 / x)))));
	} else if (t <= 1.4e-216) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 8.8e+73) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = (Math.sqrt(2.0) * Math.sqrt(0.5)) * Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -3.6e-151:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt((2.0 + (4.0 / x)))))
	elif t <= 1.4e-216:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))))
	elif t <= 2.55e-147:
		tmp = 1.0
	elif t <= 8.8e+73:
		tmp = t * (math.sqrt(2.0) / math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))))
	else:
		tmp = (math.sqrt(2.0) * math.sqrt(0.5)) * math.sqrt(((-1.0 + x) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -3.6e-151)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(2.0 + Float64(4.0 / x)))))));
	elseif (t <= 1.4e-216)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x)))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 8.8e+73)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(Float64(t * Float64(t * 2.0)) / x) + Float64(Float64(t * t) + Float64(Float64(l * l) / x)))))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(0.5)) * sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.6e-151)
		tmp = t * (sqrt(2.0) / (t * -sqrt((2.0 + (4.0 / x)))));
	elseif (t <= 1.4e-216)
		tmp = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 8.8e+73)
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	else
		tmp = (sqrt(2.0) * sqrt(0.5)) * sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -3.6e-151], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-216], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 8.8e+73], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -3.60000000000000032e-151

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. sub-neg 56.8%

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

      2. fma-def 56.8%

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

      3. unpow2 56.8%

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

      4. fma-def 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. mul-1-neg 56.8%

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

      8. remove-double-neg 56.8%

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

      9. fma-def 56.8%

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

      10. unpow2 56.8%

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

      11. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in t around -inf 88.8%

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

      1. mul-1-neg 88.8%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]

      2. distribute-rgt-neg-in 88.8%

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

      3. associate-*r/ 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -3.60000000000000032e-151 < t < 1.4e-216

   1. Initial program 2.1%

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

      1. associate-*l/ 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in l around inf 4.7%

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

      1. associate--l+ 26.6%

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

      2. sub-neg 26.6%

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

      3. metadata-eval 26.6%

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

      4. +-commutative 26.6%

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

      5. sub-neg 26.6%

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

      6. metadata-eval 26.6%

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

      7. +-commutative 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around inf 40.9%

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

   if 1.4e-216 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 8.8e73

   1. Initial program 58.1%

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

      1. associate-*l/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in l around 0 49.9%

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

      1. fma-def 49.9%

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

      2. +-commutative 49.9%

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

      3. associate-*r/ 69.2%

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

      4. sub-neg 69.2%

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

      5. metadata-eval 69.2%

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

      6. unpow2 69.2%

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

      7. +-commutative 69.2%

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

      8. unpow2 69.2%

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

      9. associate--l+ 74.4%

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

      10. sub-neg 74.4%

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

      11. metadata-eval 74.4%

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

      12. +-commutative 74.4%

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

      13. sub-neg 74.4%

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

      14. metadata-eval 74.4%

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

      15. +-commutative 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in x around inf 87.1%

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

      1. distribute-lft-out 87.1%

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

      2. distribute-lft-out 87.1%

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

      3. cancel-sign-sub-inv 87.1%

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

      4. metadata-eval 87.1%

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

      5. distribute-rgt1-in 87.1%

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

      6. metadata-eval 87.1%

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

      7. *-commutative 87.1%

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

      8. unpow2 87.1%

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

      9. associate-*l* 87.1%

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

      10. unpow2 87.1%

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

      11. unpow2 87.1%

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

   9. Simplified 87.1%

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

   if 8.8e73 < t

   1. Initial program 30.3%

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

      1. associate-*l/ 30.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in t around inf 89.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+73}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 7: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{+75}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.75e-150)
   (* t (/ (sqrt 2.0) (* t (- (sqrt (+ 2.0 (/ 4.0 x)))))))
   (if (<= t 3e-217)
     (* t (/ (sqrt 2.0) (* l (* (sqrt 2.0) (sqrt (/ 1.0 x))))))
     (if (<= t 2.55e-147)
       1.0
       (if (<= t 6.9e+75)
         (*
          t
          (/
           (sqrt 2.0)
           (sqrt (* 2.0 (+ (/ (* t (* t 2.0)) x) (+ (* t t) (/ (* l l) x)))))))
         (* (* (sqrt 2.0) (sqrt 0.5)) (sqrt (/ (+ -1.0 x) (+ x 1.0)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.75e-150) {
		tmp = t * (sqrt(2.0) / (t * -sqrt((2.0 + (4.0 / x)))));
	} else if (t <= 3e-217) {
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 6.9e+75) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = (sqrt(2.0) * sqrt(0.5)) * sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.75d-150)) then
        tmp = t * (sqrt(2.0d0) / (t * -sqrt((2.0d0 + (4.0d0 / x)))))
    else if (t <= 3d-217) then
        tmp = t * (sqrt(2.0d0) / (l * (sqrt(2.0d0) * sqrt((1.0d0 / x)))))
    else if (t <= 2.55d-147) then
        tmp = 1.0d0
    else if (t <= 6.9d+75) then
        tmp = t * (sqrt(2.0d0) / sqrt((2.0d0 * (((t * (t * 2.0d0)) / x) + ((t * t) + ((l * l) / x))))))
    else
        tmp = (sqrt(2.0d0) * sqrt(0.5d0)) * sqrt((((-1.0d0) + x) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.75e-150) {
		tmp = t * (Math.sqrt(2.0) / (t * -Math.sqrt((2.0 + (4.0 / x)))));
	} else if (t <= 3e-217) {
		tmp = t * (Math.sqrt(2.0) / (l * (Math.sqrt(2.0) * Math.sqrt((1.0 / x)))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 6.9e+75) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = (Math.sqrt(2.0) * Math.sqrt(0.5)) * Math.sqrt(((-1.0 + x) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.75e-150:
		tmp = t * (math.sqrt(2.0) / (t * -math.sqrt((2.0 + (4.0 / x)))))
	elif t <= 3e-217:
		tmp = t * (math.sqrt(2.0) / (l * (math.sqrt(2.0) * math.sqrt((1.0 / x)))))
	elif t <= 2.55e-147:
		tmp = 1.0
	elif t <= 6.9e+75:
		tmp = t * (math.sqrt(2.0) / math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))))
	else:
		tmp = (math.sqrt(2.0) * math.sqrt(0.5)) * math.sqrt(((-1.0 + x) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.75e-150)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * Float64(-sqrt(Float64(2.0 + Float64(4.0 / x)))))));
	elseif (t <= 3e-217)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * Float64(sqrt(2.0) * sqrt(Float64(1.0 / x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 6.9e+75)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(Float64(t * Float64(t * 2.0)) / x) + Float64(Float64(t * t) + Float64(Float64(l * l) / x)))))));
	else
		tmp = Float64(Float64(sqrt(2.0) * sqrt(0.5)) * sqrt(Float64(Float64(-1.0 + x) / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.75e-150)
		tmp = t * (sqrt(2.0) / (t * -sqrt((2.0 + (4.0 / x)))));
	elseif (t <= 3e-217)
		tmp = t * (sqrt(2.0) / (l * (sqrt(2.0) * sqrt((1.0 / x)))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 6.9e+75)
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	else
		tmp = (sqrt(2.0) * sqrt(0.5)) * sqrt(((-1.0 + x) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.75e-150], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * (-N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3e-217], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 6.9e+75], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(-1.0 + x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.7499999999999999e-150

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. sub-neg 56.8%

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

      2. fma-def 56.8%

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

      3. unpow2 56.8%

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

      4. fma-def 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. mul-1-neg 56.8%

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

      8. remove-double-neg 56.8%

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

      9. fma-def 56.8%

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

      10. unpow2 56.8%

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

      11. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in t around -inf 88.8%

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

      1. mul-1-neg 88.8%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]

      2. distribute-rgt-neg-in 88.8%

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

      3. associate-*r/ 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -1.7499999999999999e-150 < t < 3.00000000000000004e-217

   1. Initial program 2.1%

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

      1. associate-*l/ 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in x around inf 56.0%

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

      1. sub-neg 56.0%

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

      2. fma-def 56.0%

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

      3. unpow2 56.0%

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

      4. fma-def 56.0%

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

      5. unpow2 56.0%

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

      6. unpow2 56.0%

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

      7. mul-1-neg 56.0%

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

      8. remove-double-neg 56.0%

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

      9. fma-def 56.0%

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

      10. unpow2 56.0%

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

      11. unpow2 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in t around 0 41.0%

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

      1. associate-*l* 41.0%

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

   9. Simplified 41.0%

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

   if 3.00000000000000004e-217 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 6.9000000000000004e75

   1. Initial program 58.1%

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

      1. associate-*l/ 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in l around 0 49.9%

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

      1. fma-def 49.9%

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

      2. +-commutative 49.9%

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

      3. associate-*r/ 69.2%

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

      4. sub-neg 69.2%

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

      5. metadata-eval 69.2%

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

      6. unpow2 69.2%

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

      7. +-commutative 69.2%

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

      8. unpow2 69.2%

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

      9. associate--l+ 74.4%

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

      10. sub-neg 74.4%

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

      11. metadata-eval 74.4%

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

      12. +-commutative 74.4%

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

      13. sub-neg 74.4%

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

      14. metadata-eval 74.4%

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

      15. +-commutative 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in x around inf 87.1%

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

      1. distribute-lft-out 87.1%

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

      2. distribute-lft-out 87.1%

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

      3. cancel-sign-sub-inv 87.1%

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

      4. metadata-eval 87.1%

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

      5. distribute-rgt1-in 87.1%

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

      6. metadata-eval 87.1%

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

      7. *-commutative 87.1%

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

      8. unpow2 87.1%

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

      9. associate-*l* 87.1%

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

      10. unpow2 87.1%

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

      11. unpow2 87.1%

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

   9. Simplified 87.1%

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

   if 6.9000000000000004e75 < t

   1. Initial program 30.3%

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

      1. associate-*l/ 30.3%

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

   3. Simplified 30.3%

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

   4. Taylor expanded in t around inf 89.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-150}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-217}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1}{x}}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6.9 \cdot 10^{+75}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{-1 + x}{x + 1}}\\ \end{array} \]

Alternative 8: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+42}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 2.0 (/ 4.0 x)))))
   (if (<= t -1.4e-151)
     (* t (/ (sqrt 2.0) (* t_1 (- t))))
     (if (<= t 8.6e-218)
       (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x)))))))
       (if (<= t 2.55e-147)
         1.0
         (if (<= t 4.7e+42)
           (*
            t
            (/
             (sqrt 2.0)
             (sqrt
              (* 2.0 (+ (/ (* t (* t 2.0)) x) (+ (* t t) (/ (* l l) x)))))))
           (* t (/ (sqrt 2.0) (* t t_1)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -1.4e-151) {
		tmp = t * (sqrt(2.0) / (t_1 * -t));
	} else if (t <= 8.6e-218) {
		tmp = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 4.7e+42) {
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = t * (sqrt(2.0) / (t * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 + (4.0d0 / x)))
    if (t <= (-1.4d-151)) then
        tmp = t * (sqrt(2.0d0) / (t_1 * -t))
    else if (t <= 8.6d-218) then
        tmp = t * (sqrt(2.0d0) / (l * sqrt(((1.0d0 / x) + (1.0d0 / ((-1.0d0) + x))))))
    else if (t <= 2.55d-147) then
        tmp = 1.0d0
    else if (t <= 4.7d+42) then
        tmp = t * (sqrt(2.0d0) / sqrt((2.0d0 * (((t * (t * 2.0d0)) / x) + ((t * t) + ((l * l) / x))))))
    else
        tmp = t * (sqrt(2.0d0) / (t * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -1.4e-151) {
		tmp = t * (Math.sqrt(2.0) / (t_1 * -t));
	} else if (t <= 8.6e-218) {
		tmp = t * (Math.sqrt(2.0) / (l * Math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	} else if (t <= 2.55e-147) {
		tmp = 1.0;
	} else if (t <= 4.7e+42) {
		tmp = t * (Math.sqrt(2.0) / Math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	} else {
		tmp = t * (Math.sqrt(2.0) / (t * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt((2.0 + (4.0 / x)))
	tmp = 0
	if t <= -1.4e-151:
		tmp = t * (math.sqrt(2.0) / (t_1 * -t))
	elif t <= 8.6e-218:
		tmp = t * (math.sqrt(2.0) / (l * math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))))
	elif t <= 2.55e-147:
		tmp = 1.0
	elif t <= 4.7e+42:
		tmp = t * (math.sqrt(2.0) / math.sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))))
	else:
		tmp = t * (math.sqrt(2.0) / (t * t_1))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(2.0 + Float64(4.0 / x)))
	tmp = 0.0
	if (t <= -1.4e-151)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * Float64(-t))));
	elseif (t <= 8.6e-218)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x)))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 4.7e+42)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(2.0 * Float64(Float64(Float64(t * Float64(t * 2.0)) / x) + Float64(Float64(t * t) + Float64(Float64(l * l) / x)))))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt((2.0 + (4.0 / x)));
	tmp = 0.0;
	if (t <= -1.4e-151)
		tmp = t * (sqrt(2.0) / (t_1 * -t));
	elseif (t <= 8.6e-218)
		tmp = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	elseif (t <= 2.55e-147)
		tmp = 1.0;
	elseif (t <= 4.7e+42)
		tmp = t * (sqrt(2.0) / sqrt((2.0 * (((t * (t * 2.0)) / x) + ((t * t) + ((l * l) / x))))));
	else
		tmp = t * (sqrt(2.0) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.4e-151], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.6e-218], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.55e-147], 1.0, If[LessEqual[t, 4.7e+42], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 * N[(N[(N[(t * N[(t * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.4e-151

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. sub-neg 56.8%

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

      2. fma-def 56.8%

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

      3. unpow2 56.8%

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

      4. fma-def 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. mul-1-neg 56.8%

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

      8. remove-double-neg 56.8%

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

      9. fma-def 56.8%

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

      10. unpow2 56.8%

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

      11. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in t around -inf 88.8%

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

      1. mul-1-neg 88.8%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]

      2. distribute-rgt-neg-in 88.8%

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

      3. associate-*r/ 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -1.4e-151 < t < 8.6e-218

   1. Initial program 2.1%

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

      1. associate-*l/ 2.1%

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

   3. Simplified 2.1%

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

   4. Taylor expanded in l around inf 4.7%

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

      1. associate--l+ 26.6%

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

      2. sub-neg 26.6%

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

      3. metadata-eval 26.6%

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

      4. +-commutative 26.6%

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

      5. sub-neg 26.6%

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

      6. metadata-eval 26.6%

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

      7. +-commutative 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in x around inf 40.9%

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

   if 8.6e-218 < t < 2.5499999999999998e-147

   1. Initial program 24.9%

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

      1. associate-*l/ 24.9%

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

   3. Simplified 24.9%

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

      1. expm1-log1p-u 23.2%

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

      2. expm1-udef 23.2%

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

   5. Applied egg-rr 17.1%

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

      1. expm1-def 17.1%

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

      2. expm1-log1p 18.5%

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

      3. metadata-eval 18.5%

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

      4. sub-neg 18.5%

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

      5. associate-/l* 2.0%

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

      6. sub-neg 2.0%

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

      7. metadata-eval 2.0%

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

      8. +-commutative 2.0%

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

   7. Simplified 2.0%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. unpow2 17.7%

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

   10. Simplified 17.7%

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

   11. Taylor expanded in t around 0 70.9%

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

   if 2.5499999999999998e-147 < t < 4.69999999999999986e42

   1. Initial program 50.5%

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

      1. associate-*l/ 50.6%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in l around 0 53.5%

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

      1. fma-def 53.5%

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

      2. +-commutative 53.5%

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

      3. associate-*r/ 61.6%

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

      4. sub-neg 61.6%

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

      5. metadata-eval 61.6%

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

      6. unpow2 61.6%

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

      7. +-commutative 61.6%

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

      8. unpow2 61.6%

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

      9. associate--l+ 68.1%

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

      10. sub-neg 68.1%

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

      11. metadata-eval 68.1%

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

      12. +-commutative 68.1%

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

      13. sub-neg 68.1%

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

      14. metadata-eval 68.1%

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

      15. +-commutative 68.1%

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

   6. Simplified 68.1%

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

   7. Taylor expanded in x around inf 84.0%

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

      1. distribute-lft-out 84.0%

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

      2. distribute-lft-out 84.0%

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

      3. cancel-sign-sub-inv 84.0%

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

      4. metadata-eval 84.0%

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

      5. distribute-rgt1-in 84.0%

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

      6. metadata-eval 84.0%

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

      7. *-commutative 84.0%

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

      8. unpow2 84.0%

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

      9. associate-*l* 84.0%

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

      10. unpow2 84.0%

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

      11. unpow2 84.0%

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

   9. Simplified 84.0%

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

   if 4.69999999999999986e42 < t

   1. Initial program 37.7%

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

      1. associate-*l/ 37.7%

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

   3. Simplified 37.7%

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

   4. Taylor expanded in x around inf 39.7%

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

      1. sub-neg 39.7%

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

      2. fma-def 39.7%

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

      3. unpow2 39.7%

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

      4. fma-def 39.7%

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

      5. unpow2 39.7%

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

      6. unpow2 39.7%

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

      7. mul-1-neg 39.7%

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

      8. remove-double-neg 39.7%

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

      9. fma-def 39.7%

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

      10. unpow2 39.7%

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

      11. unpow2 39.7%

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

   6. Simplified 39.7%

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

   7. Taylor expanded in t around inf 90.4%

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

      1. associate-*r/ 90.4%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}} \cdot t \]

      2. metadata-eval 90.4%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{\color{blue}{4}}{x}}} \cdot t \]

   9. Simplified 90.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + \frac{4}{x}}}} \cdot t \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-218}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+42}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\frac{t \cdot \left(t \cdot 2\right)}{x} + \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 9: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ t_2 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_2 \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-216}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{-93}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (* t (/ (sqrt 2.0) (* l (sqrt (+ (/ 1.0 x) (/ 1.0 (+ -1.0 x))))))))
        (t_2 (sqrt (+ 2.0 (/ 4.0 x)))))
   (if (<= t -2.9e-151)
     (* t (/ (sqrt 2.0) (* t_2 (- t))))
     (if (<= t 1.15e-216)
       t_1
       (if (<= t 9.6e-147)
         1.0
         (if (<= t 1e-93) t_1 (* t (/ (sqrt 2.0) (* t t_2)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	double t_2 = sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -2.9e-151) {
		tmp = t * (sqrt(2.0) / (t_2 * -t));
	} else if (t <= 1.15e-216) {
		tmp = t_1;
	} else if (t <= 9.6e-147) {
		tmp = 1.0;
	} else if (t <= 1e-93) {
		tmp = t_1;
	} else {
		tmp = t * (sqrt(2.0) / (t * t_2));
	}
	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) :: tmp
    t_1 = t * (sqrt(2.0d0) / (l * sqrt(((1.0d0 / x) + (1.0d0 / ((-1.0d0) + x))))))
    t_2 = sqrt((2.0d0 + (4.0d0 / x)))
    if (t <= (-2.9d-151)) then
        tmp = t * (sqrt(2.0d0) / (t_2 * -t))
    else if (t <= 1.15d-216) then
        tmp = t_1
    else if (t <= 9.6d-147) then
        tmp = 1.0d0
    else if (t <= 1d-93) then
        tmp = t_1
    else
        tmp = t * (sqrt(2.0d0) / (t * t_2))
    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 * Math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	double t_2 = Math.sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -2.9e-151) {
		tmp = t * (Math.sqrt(2.0) / (t_2 * -t));
	} else if (t <= 1.15e-216) {
		tmp = t_1;
	} else if (t <= 9.6e-147) {
		tmp = 1.0;
	} else if (t <= 1e-93) {
		tmp = t_1;
	} else {
		tmp = t * (Math.sqrt(2.0) / (t * t_2));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = t * (math.sqrt(2.0) / (l * math.sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))))
	t_2 = math.sqrt((2.0 + (4.0 / x)))
	tmp = 0
	if t <= -2.9e-151:
		tmp = t * (math.sqrt(2.0) / (t_2 * -t))
	elif t <= 1.15e-216:
		tmp = t_1
	elif t <= 9.6e-147:
		tmp = 1.0
	elif t <= 1e-93:
		tmp = t_1
	else:
		tmp = t * (math.sqrt(2.0) / (t * t_2))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * Float64(sqrt(2.0) / Float64(l * sqrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(-1.0 + x)))))))
	t_2 = sqrt(Float64(2.0 + Float64(4.0 / x)))
	tmp = 0.0
	if (t <= -2.9e-151)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_2 * Float64(-t))));
	elseif (t <= 1.15e-216)
		tmp = t_1;
	elseif (t <= 9.6e-147)
		tmp = 1.0;
	elseif (t <= 1e-93)
		tmp = t_1;
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * t_2)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = t * (sqrt(2.0) / (l * sqrt(((1.0 / x) + (1.0 / (-1.0 + x))))));
	t_2 = sqrt((2.0 + (4.0 / x)));
	tmp = 0.0;
	if (t <= -2.9e-151)
		tmp = t * (sqrt(2.0) / (t_2 * -t));
	elseif (t <= 1.15e-216)
		tmp = t_1;
	elseif (t <= 9.6e-147)
		tmp = 1.0;
	elseif (t <= 1e-93)
		tmp = t_1;
	else
		tmp = t * (sqrt(2.0) / (t * t_2));
	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[(N[Sqrt[2.0], $MachinePrecision] / N[(l * N[Sqrt[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.9e-151], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$2 * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-216], t$95$1, If[LessEqual[t, 9.6e-147], 1.0, If[LessEqual[t, 1e-93], t$95$1, N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.90000000000000013e-151

   1. Initial program 39.0%

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

      1. associate-*l/ 39.0%

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

   3. Simplified 39.0%

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

   4. Taylor expanded in x around inf 56.8%

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

      1. sub-neg 56.8%

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

      2. fma-def 56.8%

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

      3. unpow2 56.8%

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

      4. fma-def 56.8%

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

      5. unpow2 56.8%

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

      6. unpow2 56.8%

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

      7. mul-1-neg 56.8%

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

      8. remove-double-neg 56.8%

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

      9. fma-def 56.8%

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

      10. unpow2 56.8%

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

      11. unpow2 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in t around -inf 88.8%

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

      1. mul-1-neg 88.8%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]

      2. distribute-rgt-neg-in 88.8%

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

      3. associate-*r/ 88.8%

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

      4. metadata-eval 88.8%

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

   9. Simplified 88.8%

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

   if -2.90000000000000013e-151 < t < 1.14999999999999998e-216 or 9.59999999999999994e-147 < t < 9.999999999999999e-94

   1. Initial program 1.9%

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

      1. associate-*l/ 1.9%

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

   3. Simplified 1.9%

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

   4. Taylor expanded in l around inf 4.1%

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

      1. associate--l+ 26.4%

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

      2. sub-neg 26.4%

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

      3. metadata-eval 26.4%

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

      4. +-commutative 26.4%

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

      5. sub-neg 26.4%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \left(\frac{x}{\color{blue}{x + \left(-1\right)}} - 1\right)}} \cdot t \]

      6. metadata-eval 26.4%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \left(\frac{x}{x + \color{blue}{-1}} - 1\right)}} \cdot t \]

      7. +-commutative 26.4%

         \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \left(\frac{x}{\color{blue}{-1 + x}} - 1\right)}} \cdot t \]

   6. Simplified 26.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \left(\frac{x}{-1 + x} - 1\right)}}} \cdot t \]

   7. Taylor expanded in x around inf 47.6%

      \[\leadsto \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{-1 + x} + \color{blue}{\frac{1}{x}}}} \cdot t \]

   if 1.14999999999999998e-216 < t < 9.59999999999999994e-147

   1. Initial program 24.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 24.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 24.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 23.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 17.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 17.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 18.5%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

      3. metadata-eval 18.5%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 18.5%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      5. associate-/l* 2.0%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg 2.0%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      7. metadata-eval 2.0%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      8. +-commutative 2.0%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. Simplified 2.0%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

   8. Taylor expanded in x around inf 17.7%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot {t}^{2}}}} \cdot t \]
   9. Step-by-step derivation

      1. unpow2 17.7%

         \[\leadsto \sqrt{\frac{2}{2 \cdot \color{blue}{\left(t \cdot t\right)}}} \cdot t \]

   10. Simplified 17.7%

       \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot \left(t \cdot t\right)}}} \cdot t \]

   11. Taylor expanded in t around 0 70.9%

       \[\leadsto \color{blue}{1} \]

   if 9.999999999999999e-94 < t

   1. Initial program 45.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 45.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 45.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 53.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]
   5. Step-by-step derivation

      1. sub-neg 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}}} \cdot t \]

      2. fma-def 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)} + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      3. unpow2 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      4. fma-def 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \color{blue}{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \color{blue}{t \cdot t}, \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      6. unpow2 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      7. mul-1-neg 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \left(-\color{blue}{\left(-\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      8. remove-double-neg 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \color{blue}{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]

      9. fma-def 53.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}}{x}}} \cdot t \]

      10. unpow2 53.7%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, {\ell}^{2}\right)}{x}}} \cdot t \]

      11. unpow2 53.7%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \color{blue}{\ell \cdot \ell}\right)}{x}}} \cdot t \]

   6. Simplified 53.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}} \cdot t \]

   7. Taylor expanded in t around inf 87.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. associate-*r/ 87.1%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}} \cdot t \]

      2. metadata-eval 87.1%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{\color{blue}{4}}{x}}} \cdot t \]

   9. Simplified 87.1%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + \frac{4}{x}}}} \cdot t \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-151}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-216}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-147}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 10^{-93}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\ell \cdot \sqrt{\frac{1}{x} + \frac{1}{-1 + x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 10: 76.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{2 + \frac{4}{x}}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{-295}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot t_1}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (+ 2.0 (/ 4.0 x)))))
   (if (<= t -1.65e-295)
     (* t (/ (sqrt 2.0) (* t_1 (- t))))
     (* t (/ (sqrt 2.0) (* t t_1))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -1.65e-295) {
		tmp = t * (sqrt(2.0) / (t_1 * -t));
	} else {
		tmp = t * (sqrt(2.0) / (t * t_1));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((2.0d0 + (4.0d0 / x)))
    if (t <= (-1.65d-295)) then
        tmp = t * (sqrt(2.0d0) / (t_1 * -t))
    else
        tmp = t * (sqrt(2.0d0) / (t * t_1))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt((2.0 + (4.0 / x)));
	double tmp;
	if (t <= -1.65e-295) {
		tmp = t * (Math.sqrt(2.0) / (t_1 * -t));
	} else {
		tmp = t * (Math.sqrt(2.0) / (t * t_1));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt((2.0 + (4.0 / x)))
	tmp = 0
	if t <= -1.65e-295:
		tmp = t * (math.sqrt(2.0) / (t_1 * -t))
	else:
		tmp = t * (math.sqrt(2.0) / (t * t_1))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(2.0 + Float64(4.0 / x)))
	tmp = 0.0
	if (t <= -1.65e-295)
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * Float64(-t))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * t_1)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt((2.0 + (4.0 / x)));
	tmp = 0.0;
	if (t <= -1.65e-295)
		tmp = t * (sqrt(2.0) / (t_1 * -t));
	else
		tmp = t * (sqrt(2.0) / (t * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.65e-295], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6499999999999999e-295

   1. Initial program 33.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 33.5%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 33.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 56.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]
   5. Step-by-step derivation

      1. sub-neg 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}}} \cdot t \]

      2. fma-def 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)} + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      3. unpow2 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      4. fma-def 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \color{blue}{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \color{blue}{t \cdot t}, \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      6. unpow2 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      7. mul-1-neg 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \left(-\color{blue}{\left(-\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      8. remove-double-neg 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \color{blue}{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]

      9. fma-def 56.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}}{x}}} \cdot t \]

      10. unpow2 56.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, {\ell}^{2}\right)}{x}}} \cdot t \]

      11. unpow2 56.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \color{blue}{\ell \cdot \ell}\right)}{x}}} \cdot t \]

   6. Simplified 56.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}} \cdot t \]

   7. Taylor expanded in t around -inf 81.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}\right)}} \cdot t \]
   8. Step-by-step derivation

      1. mul-1-neg 81.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{-t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]

      2. distribute-rgt-neg-in 81.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \left(-\sqrt{2 + 4 \cdot \frac{1}{x}}\right)}} \cdot t \]

      3. associate-*r/ 81.4%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}\right)} \cdot t \]

      4. metadata-eval 81.4%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \left(-\sqrt{2 + \frac{\color{blue}{4}}{x}}\right)} \cdot t \]

   9. Simplified 81.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \left(-\sqrt{2 + \frac{4}{x}}\right)}} \cdot t \]

   if -1.6499999999999999e-295 < t

   1. Initial program 35.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 35.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 35.9%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 54.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]
   5. Step-by-step derivation

      1. sub-neg 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}}} \cdot t \]

      2. fma-def 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)} + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      3. unpow2 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      4. fma-def 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \color{blue}{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \color{blue}{t \cdot t}, \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      6. unpow2 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      7. mul-1-neg 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \left(-\color{blue}{\left(-\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      8. remove-double-neg 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \color{blue}{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]

      9. fma-def 54.7%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}}{x}}} \cdot t \]

      10. unpow2 54.7%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, {\ell}^{2}\right)}{x}}} \cdot t \]

      11. unpow2 54.7%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \color{blue}{\ell \cdot \ell}\right)}{x}}} \cdot t \]

   6. Simplified 54.7%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}} \cdot t \]

   7. Taylor expanded in t around inf 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. associate-*r/ 73.6%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}} \cdot t \]

      2. metadata-eval 73.6%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{\color{blue}{4}}{x}}} \cdot t \]

   9. Simplified 73.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + \frac{4}{x}}}} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-295}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}} \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 11: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t 5e-309) -1.0 (* t (/ (sqrt 2.0) (* t (sqrt (+ 2.0 (/ 4.0 x))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= 5e-309) {
		tmp = -1.0;
	} else {
		tmp = t * (sqrt(2.0) / (t * sqrt((2.0 + (4.0 / 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-309) then
        tmp = -1.0d0
    else
        tmp = t * (sqrt(2.0d0) / (t * sqrt((2.0d0 + (4.0d0 / 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-309) {
		tmp = -1.0;
	} else {
		tmp = t * (Math.sqrt(2.0) / (t * Math.sqrt((2.0 + (4.0 / x)))));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= 5e-309:
		tmp = -1.0
	else:
		tmp = t * (math.sqrt(2.0) / (t * math.sqrt((2.0 + (4.0 / x)))))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= 5e-309)
		tmp = -1.0;
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t * sqrt(Float64(2.0 + Float64(4.0 / x))))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= 5e-309)
		tmp = -1.0;
	else
		tmp = t * (sqrt(2.0) / (t * sqrt((2.0 + (4.0 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, 5e-309], -1.0, N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t * N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.9999999999999995e-309

   1. Initial program 33.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 33.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 32.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 13.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 21.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 22.3%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

      3. metadata-eval 22.3%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 22.3%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      5. associate-/l* 28.3%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg 28.3%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      7. metadata-eval 28.3%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      8. +-commutative 28.3%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. Simplified 28.3%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

   8. Taylor expanded in x around inf 46.7%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot {t}^{2}}}} \cdot t \]
   9. Step-by-step derivation

      1. unpow2 46.7%

         \[\leadsto \sqrt{\frac{2}{2 \cdot \color{blue}{\left(t \cdot t\right)}}} \cdot t \]

   10. Simplified 46.7%

       \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot \left(t \cdot t\right)}}} \cdot t \]

   11. Taylor expanded in t around -inf 79.8%

       \[\leadsto \color{blue}{-1} \]

   if 4.9999999999999995e-309 < t

   1. Initial program 36.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 36.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in x around inf 54.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]
   5. Step-by-step derivation

      1. sub-neg 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}}} \cdot t \]

      2. fma-def 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)} + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      3. unpow2 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      4. fma-def 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \color{blue}{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)}\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      5. unpow2 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \color{blue}{t \cdot t}, \frac{{\ell}^{2}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      6. unpow2 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right) + \left(--1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}} \cdot t \]

      7. mul-1-neg 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \left(-\color{blue}{\left(-\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}\right)}\right)}} \cdot t \]

      8. remove-double-neg 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \color{blue}{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \cdot t \]

      9. fma-def 54.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\color{blue}{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}}{x}}} \cdot t \]

      10. unpow2 54.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, \color{blue}{t \cdot t}, {\ell}^{2}\right)}{x}}} \cdot t \]

      11. unpow2 54.0%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \color{blue}{\ell \cdot \ell}\right)}{x}}} \cdot t \]

   6. Simplified 54.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}} \cdot t \]

   7. Taylor expanded in t around inf 74.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}} \cdot t \]
   8. Step-by-step derivation

      1. associate-*r/ 74.6%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}} \cdot t \]

      2. metadata-eval 74.6%

         \[\leadsto \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{\color{blue}{4}}{x}}} \cdot t \]

   9. Simplified 74.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{t \cdot \sqrt{2 + \frac{4}{x}}}} \cdot t \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t \cdot \sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 12: 76.3% accurate, 73.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (if (<= t -2e-310) -1.0 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -2e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -2e-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 <= -2e-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, -2e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.999999999999994e-310

   1. Initial program 33.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 33.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 32.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 13.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 21.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 22.3%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

      3. metadata-eval 22.3%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 22.3%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      5. associate-/l* 28.3%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg 28.3%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      7. metadata-eval 28.3%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      8. +-commutative 28.3%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. Simplified 28.3%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

   8. Taylor expanded in x around inf 46.7%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot {t}^{2}}}} \cdot t \]
   9. Step-by-step derivation

      1. unpow2 46.7%

         \[\leadsto \sqrt{\frac{2}{2 \cdot \color{blue}{\left(t \cdot t\right)}}} \cdot t \]

   10. Simplified 46.7%

       \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot \left(t \cdot t\right)}}} \cdot t \]

   11. Taylor expanded in t around -inf 79.8%

       \[\leadsto \color{blue}{-1} \]

   if -1.999999999999994e-310 < t

   1. Initial program 36.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 36.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 35.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-udef 12.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

   5. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
   6. Step-by-step derivation

      1. expm1-def 19.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

      2. expm1-log1p 19.9%

         \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

      3. metadata-eval 19.9%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

      4. sub-neg 19.9%

         \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

      5. associate-/l* 30.6%

         \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

      6. sub-neg 30.6%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      7. metadata-eval 30.6%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

      8. +-commutative 30.6%

         \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. Simplified 30.6%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

   8. Taylor expanded in x around inf 41.4%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot {t}^{2}}}} \cdot t \]
   9. Step-by-step derivation

      1. unpow2 41.4%

         \[\leadsto \sqrt{\frac{2}{2 \cdot \color{blue}{\left(t \cdot t\right)}}} \cdot t \]

   10. Simplified 41.4%

       \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot \left(t \cdot t\right)}}} \cdot t \]

   11. Taylor expanded in t around 0 74.1%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 39.1% accurate, 225.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ -1 \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 -1.0)

C

l = abs(l);
double code(double x, double l, double t) {
	return -1.0;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = -1.0d0
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return -1.0;
}

Python

l = abs(l)
def code(x, l, t):
	return -1.0

Julia

l = abs(l)
function code(x, l, t)
	return -1.0
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = -1.0;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := -1.0

Derivation

1. Initial program 34.8%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 34.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 34.8%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]
4. Step-by-step derivation

   1. expm1-log1p-u 34.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)\right)} \cdot t \]

   2. expm1-udef 12.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]

5. Applied egg-rr 13.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)} - 1\right)} \cdot t \]
6. Step-by-step derivation

   1. expm1-def 20.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}\right)\right)} \cdot t \]

   2. expm1-log1p 21.1%

      \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + -1} - \ell \cdot \ell}}} \cdot t \]

   3. metadata-eval 21.1%

      \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x + \color{blue}{\left(-1\right)}} - \ell \cdot \ell}} \cdot t \]

   4. sub-neg 21.1%

      \[\leadsto \sqrt{\frac{2}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\color{blue}{x - 1}} - \ell \cdot \ell}} \cdot t \]

   5. associate-/l* 29.5%

      \[\leadsto \sqrt{\frac{2}{\color{blue}{\frac{x + 1}{\frac{x - 1}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}}} - \ell \cdot \ell}} \cdot t \]

   6. sub-neg 29.5%

      \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{x + \left(-1\right)}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   7. metadata-eval 29.5%

      \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{x + \color{blue}{-1}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

   8. +-commutative 29.5%

      \[\leadsto \sqrt{\frac{2}{\frac{x + 1}{\frac{\color{blue}{-1 + x}}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}} \cdot t \]

7. Simplified 29.5%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{x + 1}{\frac{-1 + x}{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}} - \ell \cdot \ell}}} \cdot t \]

8. Taylor expanded in x around inf 44.0%

   \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot {t}^{2}}}} \cdot t \]
9. Step-by-step derivation

   1. unpow2 44.0%

      \[\leadsto \sqrt{\frac{2}{2 \cdot \color{blue}{\left(t \cdot t\right)}}} \cdot t \]

10. Simplified 44.0%

    \[\leadsto \sqrt{\frac{2}{\color{blue}{2 \cdot \left(t \cdot t\right)}}} \cdot t \]

11. Taylor expanded in t around -inf 39.3%

    \[\leadsto \color{blue}{-1} \]

12. Final simplification 39.3%

    \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023279 
(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)))))