Toniolo and Linder, Equation (7)

Percentage Accurate: 32.9% → 85.5%

Time: 20.0s

Alternatives: 14

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 32.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := 2 \cdot \left(t \cdot t\right)\\ t_2 := \frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, t_1\right)\right) + \frac{\mathsf{fma}\left(\ell, \ell, t_1\right)}{x}}{2}}}\\ t_3 := \sqrt{\frac{x + -1}{x + 1}}\\ t_4 := t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+67}:\\ \;\;\;\;-t_3\\ \mathbf{elif}\;t \leq -3.45 \cdot 10^{-158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-225}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-276}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* 2.0 (* t t)))
        (t_2
         (/
          t
          (sqrt
           (/
            (+ (fma 2.0 (/ (* t t) x) (fma l (/ l x) t_1)) (/ (fma l l t_1) x))
            2.0))))
        (t_3 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_4 (* t (/ (sqrt x) l))))
   (if (<= t -5e+67)
     (- t_3)
     (if (<= t -3.45e-158)
       t_2
       (if (<= t -1.4e-225)
         (+ -1.0 (/ 1.0 x))
         (if (<= t 2.15e-276)
           t_4
           (if (<= t 5.2e-190)
             (*
              t
              (/
               (sqrt 2.0)
               (fma
                0.5
                (/ (* 2.0 (fma 2.0 (* t t) (* l l))) (* (sqrt 2.0) (* t x)))
                (* t (sqrt 2.0)))))
             (if (<= t 5.6e-152) t_4 (if (<= t 8.5e+22) t_2 t_3)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 2.0 * (t * t);
	double t_2 = t / sqrt(((fma(2.0, ((t * t) / x), fma(l, (l / x), t_1)) + (fma(l, l, t_1) / x)) / 2.0));
	double t_3 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_4 = t * (sqrt(x) / l);
	double tmp;
	if (t <= -5e+67) {
		tmp = -t_3;
	} else if (t <= -3.45e-158) {
		tmp = t_2;
	} else if (t <= -1.4e-225) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 2.15e-276) {
		tmp = t_4;
	} else if (t <= 5.2e-190) {
		tmp = t * (sqrt(2.0) / fma(0.5, ((2.0 * fma(2.0, (t * t), (l * l))) / (sqrt(2.0) * (t * x))), (t * sqrt(2.0))));
	} else if (t <= 5.6e-152) {
		tmp = t_4;
	} else if (t <= 8.5e+22) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(2.0 * Float64(t * t))
	t_2 = Float64(t / sqrt(Float64(Float64(fma(2.0, Float64(Float64(t * t) / x), fma(l, Float64(l / x), t_1)) + Float64(fma(l, l, t_1) / x)) / 2.0)))
	t_3 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_4 = Float64(t * Float64(sqrt(x) / l))
	tmp = 0.0
	if (t <= -5e+67)
		tmp = Float64(-t_3);
	elseif (t <= -3.45e-158)
		tmp = t_2;
	elseif (t <= -1.4e-225)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 2.15e-276)
		tmp = t_4;
	elseif (t <= 5.2e-190)
		tmp = Float64(t * Float64(sqrt(2.0) / fma(0.5, Float64(Float64(2.0 * fma(2.0, Float64(t * t), Float64(l * l))) / Float64(sqrt(2.0) * Float64(t * x))), Float64(t * sqrt(2.0)))));
	elseif (t <= 5.6e-152)
		tmp = t_4;
	elseif (t <= 8.5e+22)
		tmp = t_2;
	else
		tmp = 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[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Sqrt[N[(N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l + t$95$1), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e+67], (-t$95$3), If[LessEqual[t, -3.45e-158], t$95$2, If[LessEqual[t, -1.4e-225], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e-276], t$95$4, If[LessEqual[t, 5.2e-190], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(0.5 * N[(N[(2.0 * N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e-152], t$95$4, If[LessEqual[t, 8.5e+22], t$95$2, t$95$3]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.99999999999999976e67

   1. Initial program 22.8%

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

      1. associate-*l/ 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in t around inf 9.5%

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

      1. +-commutative 9.5%

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

      2. associate-*r/ 27.2%

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

      3. sub-neg 27.2%

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

      4. metadata-eval 27.2%

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

      5. unpow2 27.2%

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

      6. +-commutative 27.2%

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

   6. Simplified 27.2%

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

   7. Taylor expanded in t around -inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. sub-neg 90.9%

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

      3. metadata-eval 90.9%

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

   9. Simplified 90.9%

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

   if -4.99999999999999976e67 < t < -3.4499999999999998e-158 or 5.59999999999999969e-152 < t < 8.49999999999999979e22

   1. Initial program 58.3%

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

      1. associate-*l/ 58.4%

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

   3. Simplified 58.4%

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

      1. *-commutative 58.4%

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

      2. clear-num 58.3%

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

      3. un-div-inv 58.3%

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

      4. sqrt-undiv 58.7%

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

      5. metadata-eval 58.7%

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

      6. sub-neg 58.7%

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

      7. associate-*l/ 53.7%

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

      8. sub-neg 53.7%

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

      9. metadata-eval 53.7%

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

   5. Applied egg-rr 53.7%

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

      1. metadata-eval 53.7%

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

      2. sub-neg 53.7%

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

      3. associate-/l* 36.1%

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

      4. sub-neg 36.1%

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

      5. metadata-eval 36.1%

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

      6. +-commutative 36.1%

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

   7. Simplified 36.1%

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

   8. Taylor expanded in x around inf 88.0%

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

      1. sub-neg 88.0%

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

      2. fma-def 88.0%

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

      3. unpow2 88.0%

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

      4. +-commutative 88.0%

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

      5. unpow2 88.0%

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

      6. associate-*l/ 88.0%

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

      7. *-commutative 88.0%

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

      8. fma-def 88.0%

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

      9. unpow2 88.0%

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

      10. mul-1-neg 88.0%

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

      11. remove-double-neg 88.0%

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

   10. Simplified 88.0%

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

   if -3.4499999999999998e-158 < t < -1.4e-225

   1. Initial program 4.9%

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

      1. associate-*l/ 4.9%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in t around inf 10.2%

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

      1. +-commutative 10.2%

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

      2. associate-*r/ 10.2%

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

      3. sub-neg 10.2%

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

      4. metadata-eval 10.2%

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

      5. unpow2 10.2%

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

      6. +-commutative 10.2%

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

   6. Simplified 10.2%

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

   7. Taylor expanded in t around -inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. sub-neg 84.2%

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

      3. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

   10. Taylor expanded in x around inf 84.2%

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

   if -1.4e-225 < t < 2.1499999999999998e-276 or 5.1999999999999996e-190 < t < 5.59999999999999969e-152

   1. Initial program 1.8%

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

      1. associate-*l/ 1.8%

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

   3. Simplified 1.8%

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

   4. Taylor expanded in t around 0 6.2%

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

      1. unpow2 6.2%

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

      2. +-commutative 6.2%

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

      3. sub-neg 6.2%

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

      4. metadata-eval 6.2%

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

      5. +-commutative 6.2%

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

      6. unpow2 6.2%

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

   6. Simplified 6.2%

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

   7. Taylor expanded in x around inf 55.9%

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

      1. unpow2 55.9%

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

      2. sub-neg 55.9%

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

      3. unpow2 55.9%

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

      4. neg-mul-1 55.9%

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

      5. remove-double-neg 55.9%

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

      6. distribute-lft-out 55.9%

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

   9. Simplified 55.9%

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

   10. Taylor expanded in l around 0 62.4%

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

       1. associate-*l/ 62.5%

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

       2. *-lft-identity 62.5%

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

   12. Simplified 62.5%

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

   if 2.1499999999999998e-276 < t < 5.1999999999999996e-190

   1. Initial program 2.2%

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

      1. associate-*l/ 2.2%

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

   3. Simplified 2.2%

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

   4. Taylor expanded in x around inf 82.1%

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

      1. fma-def 82.1%

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

      2. cancel-sign-sub-inv 82.1%

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

      3. metadata-eval 82.1%

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

      4. distribute-rgt1-in 82.1%

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

      5. metadata-eval 82.1%

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

      6. fma-def 82.1%

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

      7. unpow2 82.1%

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

      8. unpow2 82.1%

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

      9. associate-*r* 82.1%

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

   6. Simplified 82.1%

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

   if 8.49999999999999979e22 < t

   1. Initial program 31.3%

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

      1. associate-*l/ 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in t around inf 17.0%

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

      1. +-commutative 17.0%

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

      2. associate-*r/ 38.9%

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

      3. sub-neg 38.9%

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

      4. metadata-eval 38.9%

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

      5. unpow2 38.9%

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

      6. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in t around 0 93.4%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+67}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -3.45 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t\right)\right)\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-225}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-276}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-190}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{\sqrt{2} \cdot \left(t \cdot x\right)}, t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t\right)\right)\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 2: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := 2 \cdot \left(t \cdot t\right)\\ t_3 := \frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, t_2\right)\right) + \frac{\mathsf{fma}\left(\ell, \ell, t_2\right)}{x}}{2}}}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+68}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -1.32 \cdot 10^{-157}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-225}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+23}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0))))
        (t_2 (* 2.0 (* t t)))
        (t_3
         (/
          t
          (sqrt
           (/
            (+ (fma 2.0 (/ (* t t) x) (fma l (/ l x) t_2)) (/ (fma l l t_2) x))
            2.0)))))
   (if (<= t -6.2e+68)
     (- t_1)
     (if (<= t -1.32e-157)
       t_3
       (if (<= t -1.4e-225)
         (+ -1.0 (/ 1.0 x))
         (if (<= t 5.6e-152)
           (* t (/ (sqrt x) l))
           (if (<= t 9.5e+23) t_3 t_1)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = 2.0 * (t * t);
	double t_3 = t / sqrt(((fma(2.0, ((t * t) / x), fma(l, (l / x), t_2)) + (fma(l, l, t_2) / x)) / 2.0));
	double tmp;
	if (t <= -6.2e+68) {
		tmp = -t_1;
	} else if (t <= -1.32e-157) {
		tmp = t_3;
	} else if (t <= -1.4e-225) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 5.6e-152) {
		tmp = t * (sqrt(x) / l);
	} else if (t <= 9.5e+23) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(2.0 * Float64(t * t))
	t_3 = Float64(t / sqrt(Float64(Float64(fma(2.0, Float64(Float64(t * t) / x), fma(l, Float64(l / x), t_2)) + Float64(fma(l, l, t_2) / x)) / 2.0)))
	tmp = 0.0
	if (t <= -6.2e+68)
		tmp = Float64(-t_1);
	elseif (t <= -1.32e-157)
		tmp = t_3;
	elseif (t <= -1.4e-225)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 5.6e-152)
		tmp = Float64(t * Float64(sqrt(x) / l));
	elseif (t <= 9.5e+23)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t / N[Sqrt[N[(N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(l * N[(l / x), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l + t$95$2), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.2e+68], (-t$95$1), If[LessEqual[t, -1.32e-157], t$95$3, If[LessEqual[t, -1.4e-225], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e-152], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e+23], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.1999999999999997e68

   1. Initial program 22.8%

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

      1. associate-*l/ 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in t around inf 9.5%

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

      1. +-commutative 9.5%

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

      2. associate-*r/ 27.2%

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

      3. sub-neg 27.2%

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

      4. metadata-eval 27.2%

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

      5. unpow2 27.2%

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

      6. +-commutative 27.2%

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

   6. Simplified 27.2%

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

   7. Taylor expanded in t around -inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. sub-neg 90.9%

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

      3. metadata-eval 90.9%

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

   9. Simplified 90.9%

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

   if -6.1999999999999997e68 < t < -1.3200000000000001e-157 or 5.59999999999999969e-152 < t < 9.50000000000000038e23

   1. Initial program 58.3%

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

      1. associate-*l/ 58.4%

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

   3. Simplified 58.4%

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

      1. *-commutative 58.4%

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

      2. clear-num 58.3%

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

      3. un-div-inv 58.3%

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

      4. sqrt-undiv 58.7%

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

      5. metadata-eval 58.7%

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

      6. sub-neg 58.7%

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

      7. associate-*l/ 53.7%

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

      8. sub-neg 53.7%

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

      9. metadata-eval 53.7%

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

   5. Applied egg-rr 53.7%

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

      1. metadata-eval 53.7%

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

      2. sub-neg 53.7%

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

      3. associate-/l* 36.1%

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

      4. sub-neg 36.1%

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

      5. metadata-eval 36.1%

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

      6. +-commutative 36.1%

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

   7. Simplified 36.1%

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

   8. Taylor expanded in x around inf 88.0%

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

      1. sub-neg 88.0%

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

      2. fma-def 88.0%

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

      3. unpow2 88.0%

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

      4. +-commutative 88.0%

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

      5. unpow2 88.0%

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

      6. associate-*l/ 88.0%

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

      7. *-commutative 88.0%

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

      8. fma-def 88.0%

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

      9. unpow2 88.0%

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

      10. mul-1-neg 88.0%

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

      11. remove-double-neg 88.0%

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

   10. Simplified 88.0%

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

   if -1.3200000000000001e-157 < t < -1.4e-225

   1. Initial program 4.9%

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

      1. associate-*l/ 4.9%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in t around inf 10.2%

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

      1. +-commutative 10.2%

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

      2. associate-*r/ 10.2%

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

      3. sub-neg 10.2%

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

      4. metadata-eval 10.2%

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

      5. unpow2 10.2%

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

      6. +-commutative 10.2%

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

   6. Simplified 10.2%

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

   7. Taylor expanded in t around -inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. sub-neg 84.2%

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

      3. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

   10. Taylor expanded in x around inf 84.2%

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

   if -1.4e-225 < t < 5.59999999999999969e-152

   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 t around 0 6.9%

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

      1. unpow2 6.9%

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

      2. +-commutative 6.9%

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

      3. sub-neg 6.9%

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

      4. metadata-eval 6.9%

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

      5. +-commutative 6.9%

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

      6. unpow2 6.9%

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

   6. Simplified 6.9%

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

   7. Taylor expanded in x around inf 46.3%

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

      1. unpow2 46.3%

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

      2. sub-neg 46.3%

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

      3. unpow2 46.3%

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

      4. neg-mul-1 46.3%

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

      5. remove-double-neg 46.3%

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

      6. distribute-lft-out 46.3%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in l around 0 55.6%

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

       1. associate-*l/ 55.6%

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

       2. *-lft-identity 55.6%

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

   12. Simplified 55.6%

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

   if 9.50000000000000038e23 < t

   1. Initial program 31.3%

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

      1. associate-*l/ 31.3%

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

   3. Simplified 31.3%

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

   4. Taylor expanded in t around inf 17.0%

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

      1. +-commutative 17.0%

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

      2. associate-*r/ 38.9%

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

      3. sub-neg 38.9%

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

      4. metadata-eval 38.9%

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

      5. unpow2 38.9%

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

      6. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in t around 0 93.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{+68}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -1.32 \cdot 10^{-157}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t\right)\right)\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}}{2}}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-225}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(\ell, \frac{\ell}{x}, 2 \cdot \left(t \cdot t\right)\right)\right) + \frac{\mathsf{fma}\left(\ell, \ell, 2 \cdot \left(t \cdot t\right)\right)}{x}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 3: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ t_2 := \frac{\ell \cdot \ell}{x}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq -3.45 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{t_2 + \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, t_2\right)\right)}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-226}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot \left(t \cdot t\right)}{x + -1}, \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{x} + \frac{1}{x + -1}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))) (t_2 (/ (* l l) x)))
   (if (<= t -1.45e+67)
     (- t_1)
     (if (<= t -3.45e-158)
       (*
        t
        (/
         (sqrt 2.0)
         (sqrt (+ t_2 (fma 2.0 (/ (* t t) x) (fma 2.0 (* t t) t_2))))))
       (if (<= t -5e-226)
         (+ -1.0 (/ 1.0 x))
         (if (<= t 5.6e-152)
           (* t (/ (sqrt x) l))
           (if (<= t 1.9e-37)
             (/
              t
              (sqrt
               (/
                (fma
                 2.0
                 (/ (* (+ x 1.0) (* t t)) (+ x -1.0))
                 (* (* l l) (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))))
                2.0)))
             t_1)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double t_2 = (l * l) / x;
	double tmp;
	if (t <= -1.45e+67) {
		tmp = -t_1;
	} else if (t <= -3.45e-158) {
		tmp = t * (sqrt(2.0) / sqrt((t_2 + fma(2.0, ((t * t) / x), fma(2.0, (t * t), t_2)))));
	} else if (t <= -5e-226) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 5.6e-152) {
		tmp = t * (sqrt(x) / l);
	} else if (t <= 1.9e-37) {
		tmp = t / sqrt((fma(2.0, (((x + 1.0) * (t * t)) / (x + -1.0)), ((l * l) * ((1.0 / x) + (1.0 / (x + -1.0))))) / 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	t_2 = Float64(Float64(l * l) / x)
	tmp = 0.0
	if (t <= -1.45e+67)
		tmp = Float64(-t_1);
	elseif (t <= -3.45e-158)
		tmp = Float64(t * Float64(sqrt(2.0) / sqrt(Float64(t_2 + fma(2.0, Float64(Float64(t * t) / x), fma(2.0, Float64(t * t), t_2))))));
	elseif (t <= -5e-226)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 5.6e-152)
		tmp = Float64(t * Float64(sqrt(x) / l));
	elseif (t <= 1.9e-37)
		tmp = Float64(t / sqrt(Float64(fma(2.0, Float64(Float64(Float64(x + 1.0) * Float64(t * t)) / Float64(x + -1.0)), Float64(Float64(l * l) * Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0))))) / 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t, -1.45e+67], (-t$95$1), If[LessEqual[t, -3.45e-158], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-226], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e-152], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-37], N[(t / N[Sqrt[N[(N[(2.0 * N[(N[(N[(x + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -1.45000000000000012e67

   1. Initial program 22.8%

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

      1. associate-*l/ 22.8%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in t around inf 9.5%

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

      1. +-commutative 9.5%

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

      2. associate-*r/ 27.2%

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

      3. sub-neg 27.2%

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

      4. metadata-eval 27.2%

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

      5. unpow2 27.2%

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

      6. +-commutative 27.2%

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

   6. Simplified 27.2%

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

   7. Taylor expanded in t around -inf 90.9%

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

      1. mul-1-neg 90.9%

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

      2. sub-neg 90.9%

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

      3. metadata-eval 90.9%

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

   9. Simplified 90.9%

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

   if -1.45000000000000012e67 < t < -3.4499999999999998e-158

   1. Initial program 61.4%

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

      1. associate-*l/ 61.5%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in x around inf 92.5%

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

      1. sub-neg 92.5%

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

      2. fma-def 92.5%

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

      3. unpow2 92.5%

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

      4. fma-def 92.5%

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

      5. unpow2 92.5%

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

      6. unpow2 92.5%

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

      7. mul-1-neg 92.5%

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

      8. remove-double-neg 92.5%

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

      9. fma-def 92.5%

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

      10. unpow2 92.5%

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

      11. unpow2 92.5%

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

   6. Simplified 92.5%

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

   7. Taylor expanded in t around 0 92.1%

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

      1. unpow2 92.1%

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

   9. Simplified 92.1%

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

   if -3.4499999999999998e-158 < t < -4.9999999999999998e-226

   1. Initial program 4.9%

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

      1. associate-*l/ 4.9%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in t around inf 10.2%

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

      1. +-commutative 10.2%

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

      2. associate-*r/ 10.2%

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

      3. sub-neg 10.2%

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

      4. metadata-eval 10.2%

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

      5. unpow2 10.2%

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

      6. +-commutative 10.2%

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

   6. Simplified 10.2%

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

   7. Taylor expanded in t around -inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. sub-neg 84.2%

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

      3. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

   10. Taylor expanded in x around inf 84.2%

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

   if -4.9999999999999998e-226 < t < 5.59999999999999969e-152

   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 t around 0 6.9%

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

      1. unpow2 6.9%

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

      2. +-commutative 6.9%

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

      3. sub-neg 6.9%

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

      4. metadata-eval 6.9%

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

      5. +-commutative 6.9%

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

      6. unpow2 6.9%

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

   6. Simplified 6.9%

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

   7. Taylor expanded in x around inf 46.3%

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

      1. unpow2 46.3%

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

      2. sub-neg 46.3%

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

      3. unpow2 46.3%

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

      4. neg-mul-1 46.3%

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

      5. remove-double-neg 46.3%

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

      6. distribute-lft-out 46.3%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in l around 0 55.6%

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

       1. associate-*l/ 55.6%

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

       2. *-lft-identity 55.6%

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

   12. Simplified 55.6%

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

   if 5.59999999999999969e-152 < t < 1.9000000000000002e-37

   1. Initial program 49.0%

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

      1. associate-*l/ 49.1%

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

   3. Simplified 49.1%

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

      1. *-commutative 49.1%

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

      2. clear-num 49.0%

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

      3. un-div-inv 49.0%

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

      4. sqrt-undiv 49.2%

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

      5. metadata-eval 49.2%

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

      6. sub-neg 49.2%

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

      7. associate-*l/ 49.8%

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

      8. sub-neg 49.8%

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

      9. metadata-eval 49.8%

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

   5. Applied egg-rr 49.8%

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

      1. metadata-eval 49.8%

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

      2. sub-neg 49.8%

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

      3. associate-/l* 12.8%

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

      4. sub-neg 12.8%

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

      5. metadata-eval 12.8%

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

      6. +-commutative 12.8%

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

   7. Simplified 12.8%

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

   8. Taylor expanded in l around 0 64.0%

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

      1. fma-def 64.0%

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

      2. unpow2 64.0%

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

      3. sub-neg 64.0%

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

      4. metadata-eval 64.0%

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

      5. unpow2 64.0%

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

      6. associate--l+ 68.0%

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

      7. sub-neg 68.0%

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

      8. metadata-eval 68.0%

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

      9. sub-neg 68.0%

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

      10. metadata-eval 68.0%

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

   10. Simplified 68.0%

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

   11. Taylor expanded in x around inf 82.9%

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

   if 1.9000000000000002e-37 < t

   1. Initial program 36.1%

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

      1. associate-*l/ 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t around inf 23.7%

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

      1. +-commutative 23.7%

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

      2. associate-*r/ 44.1%

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

      3. sub-neg 44.1%

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

      4. metadata-eval 44.1%

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

      5. unpow2 44.1%

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

      6. +-commutative 44.1%

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

   6. Simplified 44.1%

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

   7. Taylor expanded in t around 0 91.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+67}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -3.45 \cdot 10^{-158}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\sqrt{\frac{\ell \cdot \ell}{x} + \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-226}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot \left(t \cdot t\right)}{x + -1}, \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{x} + \frac{1}{x + -1}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot \left(t \cdot t\right)}{x + -1}, \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{x} + \frac{1}{x + -1}\right)\right)}{2}}}\\ t_2 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;-t_2\\ \mathbf{elif}\;t \leq -3.45 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-227}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1
         (/
          t
          (sqrt
           (/
            (fma
             2.0
             (/ (* (+ x 1.0) (* t t)) (+ x -1.0))
             (* (* l l) (+ (/ 1.0 x) (/ 1.0 (+ x -1.0)))))
            2.0))))
        (t_2 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -2.7e-7)
     (- t_2)
     (if (<= t -3.45e-158)
       t_1
       (if (<= t -1.25e-227)
         (+ -1.0 (/ 1.0 x))
         (if (<= t 5.6e-152)
           (* t (/ (sqrt x) l))
           (if (<= t 1.9e-37) t_1 t_2)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t / sqrt((fma(2.0, (((x + 1.0) * (t * t)) / (x + -1.0)), ((l * l) * ((1.0 / x) + (1.0 / (x + -1.0))))) / 2.0));
	double t_2 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -2.7e-7) {
		tmp = -t_2;
	} else if (t <= -3.45e-158) {
		tmp = t_1;
	} else if (t <= -1.25e-227) {
		tmp = -1.0 + (1.0 / x);
	} else if (t <= 5.6e-152) {
		tmp = t * (sqrt(x) / l);
	} else if (t <= 1.9e-37) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t / sqrt(Float64(fma(2.0, Float64(Float64(Float64(x + 1.0) * Float64(t * t)) / Float64(x + -1.0)), Float64(Float64(l * l) * Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x + -1.0))))) / 2.0)))
	t_2 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -2.7e-7)
		tmp = Float64(-t_2);
	elseif (t <= -3.45e-158)
		tmp = t_1;
	elseif (t <= -1.25e-227)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	elseif (t <= 5.6e-152)
		tmp = Float64(t * Float64(sqrt(x) / l));
	elseif (t <= 1.9e-37)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(t / N[Sqrt[N[(N[(2.0 * N[(N[(N[(x + 1.0), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.7e-7], (-t$95$2), If[LessEqual[t, -3.45e-158], t$95$1, If[LessEqual[t, -1.25e-227], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e-152], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.9e-37], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.70000000000000009e-7

   1. Initial program 32.7%

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

      1. associate-*l/ 32.8%

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

   3. Simplified 32.8%

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

   4. Taylor expanded in t around inf 19.7%

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

      1. +-commutative 19.7%

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

      2. associate-*r/ 39.0%

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

      3. sub-neg 39.0%

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

      4. metadata-eval 39.0%

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

      5. unpow2 39.0%

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

      6. +-commutative 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in t around -inf 90.2%

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

      1. mul-1-neg 90.2%

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

      2. sub-neg 90.2%

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

      3. metadata-eval 90.2%

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

   9. Simplified 90.2%

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

   if -2.70000000000000009e-7 < t < -3.4499999999999998e-158 or 5.59999999999999969e-152 < t < 1.9000000000000002e-37

   1. Initial program 51.7%

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

      1. associate-*l/ 51.8%

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

   3. Simplified 51.8%

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

      1. *-commutative 51.8%

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

      2. clear-num 51.8%

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

      3. un-div-inv 51.7%

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

      4. sqrt-undiv 52.0%

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

      5. metadata-eval 52.0%

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

      6. sub-neg 52.0%

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

      7. associate-*l/ 52.3%

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

      8. sub-neg 52.3%

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

      9. metadata-eval 52.3%

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

   5. Applied egg-rr 52.3%

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

      1. metadata-eval 52.3%

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

      2. sub-neg 52.3%

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

      3. associate-/l* 18.7%

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

      4. sub-neg 18.7%

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

      5. metadata-eval 18.7%

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

      6. +-commutative 18.7%

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

   7. Simplified 18.7%

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

   8. Taylor expanded in l around 0 65.5%

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

      1. fma-def 65.5%

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

      2. unpow2 65.5%

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

      3. sub-neg 65.5%

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

      4. metadata-eval 65.5%

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

      5. unpow2 65.5%

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

      6. associate--l+ 71.9%

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

      7. sub-neg 71.9%

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

      8. metadata-eval 71.9%

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

      9. sub-neg 71.9%

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

      10. metadata-eval 71.9%

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

   10. Simplified 71.9%

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

   11. Taylor expanded in x around inf 88.6%

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

   if -3.4499999999999998e-158 < t < -1.2499999999999999e-227

   1. Initial program 4.9%

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

      1. associate-*l/ 4.9%

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

   3. Simplified 4.9%

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

   4. Taylor expanded in t around inf 10.2%

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

      1. +-commutative 10.2%

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

      2. associate-*r/ 10.2%

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

      3. sub-neg 10.2%

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

      4. metadata-eval 10.2%

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

      5. unpow2 10.2%

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

      6. +-commutative 10.2%

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

   6. Simplified 10.2%

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

   7. Taylor expanded in t around -inf 84.2%

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

      1. mul-1-neg 84.2%

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

      2. sub-neg 84.2%

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

      3. metadata-eval 84.2%

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

   9. Simplified 84.2%

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

   10. Taylor expanded in x around inf 84.2%

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

   if -1.2499999999999999e-227 < t < 5.59999999999999969e-152

   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 t around 0 6.9%

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

      1. unpow2 6.9%

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

      2. +-commutative 6.9%

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

      3. sub-neg 6.9%

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

      4. metadata-eval 6.9%

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

      5. +-commutative 6.9%

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

      6. unpow2 6.9%

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

   6. Simplified 6.9%

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

   7. Taylor expanded in x around inf 46.3%

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

      1. unpow2 46.3%

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

      2. sub-neg 46.3%

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

      3. unpow2 46.3%

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

      4. neg-mul-1 46.3%

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

      5. remove-double-neg 46.3%

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

      6. distribute-lft-out 46.3%

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

   9. Simplified 46.3%

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

   10. Taylor expanded in l around 0 55.6%

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

       1. associate-*l/ 55.6%

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

       2. *-lft-identity 55.6%

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

   12. Simplified 55.6%

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

   if 1.9000000000000002e-37 < t

   1. Initial program 36.1%

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

      1. associate-*l/ 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in t around inf 23.7%

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

      1. +-commutative 23.7%

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

      2. associate-*r/ 44.1%

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

      3. sub-neg 44.1%

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

      4. metadata-eval 44.1%

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

      5. unpow2 44.1%

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

      6. +-commutative 44.1%

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

   6. Simplified 44.1%

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

   7. Taylor expanded in t around 0 91.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq -3.45 \cdot 10^{-158}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot \left(t \cdot t\right)}{x + -1}, \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{x} + \frac{1}{x + -1}\right)\right)}{2}}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-227}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-152}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{t}{\sqrt{\frac{\mathsf{fma}\left(2, \frac{\left(x + 1\right) \cdot \left(t \cdot t\right)}{x + -1}, \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{x} + \frac{1}{x + -1}\right)\right)}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 5: 80.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.3e-227)
   (- (/ 1.0 x) (+ 1.0 (/ 0.5 (* x x))))
   (if (<= t 1.05e-127) (* t (/ (sqrt x) l)) (sqrt (/ (+ x -1.0) (+ x 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.3e-227) {
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)));
	} else if (t <= 1.05e-127) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.3d-227)) then
        tmp = (1.0d0 / x) - (1.0d0 + (0.5d0 / (x * x)))
    else if (t <= 1.05d-127) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.3e-227) {
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)));
	} else if (t <= 1.05e-127) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.3e-227:
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)))
	elif t <= 1.05e-127:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.3e-227)
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 + Float64(0.5 / Float64(x * x))));
	elseif (t <= 1.05e-127)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.3e-227)
		tmp = (1.0 / x) - (1.0 + (0.5 / (x * x)));
	elseif (t <= 1.05e-127)
		tmp = t * (sqrt(x) / l);
	else
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.3e-227], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-127], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.30000000000000006e-227

   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. Taylor expanded in t around inf 29.5%

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

      1. +-commutative 29.5%

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

      2. associate-*r/ 42.4%

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

      3. sub-neg 42.4%

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

      4. metadata-eval 42.4%

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

      5. unpow2 42.4%

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

      6. +-commutative 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in t around -inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in x around inf 83.6%

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

       1. associate-*r/ 83.6%

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

       2. metadata-eval 83.6%

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

       3. unpow2 83.6%

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

   12. Simplified 83.6%

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

   if -1.30000000000000006e-227 < t < 1.05000000000000005e-127

   1. Initial program 4.3%

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

      1. associate-*l/ 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in t around 0 6.3%

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

      1. unpow2 6.3%

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

      2. +-commutative 6.3%

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

      3. sub-neg 6.3%

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

      4. metadata-eval 6.3%

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

      5. +-commutative 6.3%

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

      6. unpow2 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in x around inf 43.5%

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

      1. unpow2 43.5%

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

      2. sub-neg 43.5%

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

      3. unpow2 43.5%

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

      4. neg-mul-1 43.5%

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

      5. remove-double-neg 43.5%

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

      6. distribute-lft-out 43.5%

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

   9. Simplified 43.5%

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

   10. Taylor expanded in l around 0 56.1%

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

       1. associate-*l/ 56.2%

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

       2. *-lft-identity 56.2%

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

   12. Simplified 56.2%

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

   if 1.05000000000000005e-127 < t

   1. Initial program 40.1%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 32.3%

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

      1. +-commutative 32.3%

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

      2. associate-*r/ 48.5%

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

      3. sub-neg 48.5%

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

      4. metadata-eval 48.5%

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

      5. unpow2 48.5%

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

      6. +-commutative 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in t around 0 86.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 6: 80.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-229}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (sqrt (/ (+ x -1.0) (+ x 1.0)))))
   (if (<= t -5e-229) (- t_1) (if (<= t 1.5e-127) (* t (/ (sqrt x) l)) t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -5e-229) {
		tmp = -t_1;
	} else if (t <= 1.5e-127) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    if (t <= (-5d-229)) then
        tmp = -t_1
    else if (t <= 1.5d-127) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(((x + -1.0) / (x + 1.0)));
	double tmp;
	if (t <= -5e-229) {
		tmp = -t_1;
	} else if (t <= 1.5e-127) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(((x + -1.0) / (x + 1.0)))
	tmp = 0
	if t <= -5e-229:
		tmp = -t_1
	elif t <= 1.5e-127:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)))
	tmp = 0.0
	if (t <= -5e-229)
		tmp = Float64(-t_1);
	elseif (t <= 1.5e-127)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(((x + -1.0) / (x + 1.0)));
	tmp = 0.0;
	if (t <= -5e-229)
		tmp = -t_1;
	elseif (t <= 1.5e-127)
		tmp = t * (sqrt(x) / l);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -5e-229], (-t$95$1), If[LessEqual[t, 1.5e-127], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if t < -5.00000000000000016e-229

   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. Taylor expanded in t around inf 29.5%

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

      1. +-commutative 29.5%

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

      2. associate-*r/ 42.4%

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

      3. sub-neg 42.4%

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

      4. metadata-eval 42.4%

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

      5. unpow2 42.4%

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

      6. +-commutative 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in t around -inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

   if -5.00000000000000016e-229 < t < 1.50000000000000004e-127

   1. Initial program 4.3%

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

      1. associate-*l/ 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in t around 0 6.3%

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

      1. unpow2 6.3%

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

      2. +-commutative 6.3%

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

      3. sub-neg 6.3%

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

      4. metadata-eval 6.3%

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

      5. +-commutative 6.3%

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

      6. unpow2 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in x around inf 43.5%

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

      1. unpow2 43.5%

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

      2. sub-neg 43.5%

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

      3. unpow2 43.5%

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

      4. neg-mul-1 43.5%

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

      5. remove-double-neg 43.5%

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

      6. distribute-lft-out 43.5%

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

   9. Simplified 43.5%

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

   10. Taylor expanded in l around 0 56.1%

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

       1. associate-*l/ 56.2%

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

       2. *-lft-identity 56.2%

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

   12. Simplified 56.2%

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

   if 1.50000000000000004e-127 < t

   1. Initial program 40.1%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 32.3%

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

      1. +-commutative 32.3%

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

      2. associate-*r/ 48.5%

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

      3. sub-neg 48.5%

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

      4. metadata-eval 48.5%

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

      5. unpow2 48.5%

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

      6. +-commutative 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in t around 0 86.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-229}:\\ \;\;\;\;-\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \end{array} \]

Alternative 7: 79.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{x} - \left(1 + t_1\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -8.5e-228)
     (- (/ 1.0 x) (+ 1.0 t_1))
     (if (<= t 1.05e-216) (* (sqrt x) (/ t l)) (+ 1.0 (+ t_1 (/ -1.0 x)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -8.5e-228) {
		tmp = (1.0 / x) - (1.0 + t_1);
	} else if (t <= 1.05e-216) {
		tmp = sqrt(x) * (t / l);
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-8.5d-228)) then
        tmp = (1.0d0 / x) - (1.0d0 + t_1)
    else if (t <= 1.05d-216) then
        tmp = sqrt(x) * (t / l)
    else
        tmp = 1.0d0 + (t_1 + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -8.5e-228) {
		tmp = (1.0 / x) - (1.0 + t_1);
	} else if (t <= 1.05e-216) {
		tmp = Math.sqrt(x) * (t / l);
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -8.5e-228:
		tmp = (1.0 / x) - (1.0 + t_1)
	elif t <= 1.05e-216:
		tmp = math.sqrt(x) * (t / l)
	else:
		tmp = 1.0 + (t_1 + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -8.5e-228)
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 + t_1));
	elseif (t <= 1.05e-216)
		tmp = Float64(sqrt(x) * Float64(t / l));
	else
		tmp = Float64(1.0 + Float64(t_1 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -8.5e-228)
		tmp = (1.0 / x) - (1.0 + t_1);
	elseif (t <= 1.05e-216)
		tmp = sqrt(x) * (t / l);
	else
		tmp = 1.0 + (t_1 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e-228], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-216], N[(N[Sqrt[x], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.49999999999999947e-228

   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. Taylor expanded in t around inf 29.5%

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

      1. +-commutative 29.5%

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

      2. associate-*r/ 42.4%

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

      3. sub-neg 42.4%

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

      4. metadata-eval 42.4%

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

      5. unpow2 42.4%

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

      6. +-commutative 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in t around -inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in x around inf 83.6%

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

       1. associate-*r/ 83.6%

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

       2. metadata-eval 83.6%

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

       3. unpow2 83.6%

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

   12. Simplified 83.6%

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

   if -8.49999999999999947e-228 < t < 1.0500000000000001e-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. Step-by-step derivation

      1. *-commutative 1.9%

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

      2. clear-num 1.9%

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

      3. un-div-inv 1.9%

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

      4. sqrt-undiv 1.9%

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

      5. metadata-eval 1.9%

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

      6. sub-neg 1.9%

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

      7. associate-*l/ 9.7%

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

      8. sub-neg 9.7%

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

      9. metadata-eval 9.7%

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

   5. Applied egg-rr 9.7%

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

      1. metadata-eval 9.7%

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

      2. sub-neg 9.7%

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

      3. associate-/l* 1.4%

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

      4. sub-neg 1.4%

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

      5. metadata-eval 1.4%

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

      6. +-commutative 1.4%

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

   7. Simplified 1.4%

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

   8. Taylor expanded in x around inf 55.7%

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

      1. sub-neg 55.7%

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

      2. fma-def 55.7%

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

      3. unpow2 55.7%

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

      4. +-commutative 55.7%

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

      5. unpow2 55.7%

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

      6. associate-*l/ 55.7%

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

      7. *-commutative 55.7%

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

      8. fma-def 55.6%

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

      9. unpow2 55.6%

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

      10. mul-1-neg 55.6%

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

      11. remove-double-neg 55.6%

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

   10. Simplified 55.6%

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

   11. Taylor expanded in t around 0 65.4%

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

   if 1.0500000000000001e-216 < t

   1. Initial program 34.9%

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

      1. associate-*l/ 34.9%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in t around inf 29.3%

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

      1. +-commutative 29.3%

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

      2. associate-*r/ 42.9%

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

      3. sub-neg 42.9%

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

      4. metadata-eval 42.9%

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

      5. unpow2 42.9%

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

      6. +-commutative 42.9%

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

   6. Simplified 42.9%

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

   7. Taylor expanded in x around inf 79.9%

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

      1. associate--l+ 79.9%

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

      2. associate-*r/ 79.9%

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

      3. metadata-eval 79.9%

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

      4. unpow2 79.9%

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

   9. Simplified 79.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{x} \cdot \frac{t}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 8: 80.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{x} - \left(1 + t_1\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -1.1e-226)
     (- (/ 1.0 x) (+ 1.0 t_1))
     (if (<= t 1.05e-127) (* t (/ (sqrt x) l)) (+ 1.0 (+ t_1 (/ -1.0 x)))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1.1e-226) {
		tmp = (1.0 / x) - (1.0 + t_1);
	} else if (t <= 1.05e-127) {
		tmp = t * (sqrt(x) / l);
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-1.1d-226)) then
        tmp = (1.0d0 / x) - (1.0d0 + t_1)
    else if (t <= 1.05d-127) then
        tmp = t * (sqrt(x) / l)
    else
        tmp = 1.0d0 + (t_1 + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -1.1e-226) {
		tmp = (1.0 / x) - (1.0 + t_1);
	} else if (t <= 1.05e-127) {
		tmp = t * (Math.sqrt(x) / l);
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -1.1e-226:
		tmp = (1.0 / x) - (1.0 + t_1)
	elif t <= 1.05e-127:
		tmp = t * (math.sqrt(x) / l)
	else:
		tmp = 1.0 + (t_1 + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -1.1e-226)
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 + t_1));
	elseif (t <= 1.05e-127)
		tmp = Float64(t * Float64(sqrt(x) / l));
	else
		tmp = Float64(1.0 + Float64(t_1 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -1.1e-226)
		tmp = (1.0 / x) - (1.0 + t_1);
	elseif (t <= 1.05e-127)
		tmp = t * (sqrt(x) / l);
	else
		tmp = 1.0 + (t_1 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.1e-226], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.05e-127], N[(t * N[(N[Sqrt[x], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1e-226

   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. Taylor expanded in t around inf 29.5%

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

      1. +-commutative 29.5%

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

      2. associate-*r/ 42.4%

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

      3. sub-neg 42.4%

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

      4. metadata-eval 42.4%

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

      5. unpow2 42.4%

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

      6. +-commutative 42.4%

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

   6. Simplified 42.4%

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

   7. Taylor expanded in t around -inf 84.4%

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

      1. mul-1-neg 84.4%

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

      2. sub-neg 84.4%

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

      3. metadata-eval 84.4%

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

   9. Simplified 84.4%

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

   10. Taylor expanded in x around inf 83.6%

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

       1. associate-*r/ 83.6%

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

       2. metadata-eval 83.6%

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

       3. unpow2 83.6%

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

   12. Simplified 83.6%

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

   if -1.1e-226 < t < 1.05000000000000005e-127

   1. Initial program 4.3%

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

      1. associate-*l/ 4.3%

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

   3. Simplified 4.3%

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

   4. Taylor expanded in t around 0 6.3%

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

      1. unpow2 6.3%

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

      2. +-commutative 6.3%

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

      3. sub-neg 6.3%

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

      4. metadata-eval 6.3%

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

      5. +-commutative 6.3%

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

      6. unpow2 6.3%

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

   6. Simplified 6.3%

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

   7. Taylor expanded in x around inf 43.5%

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

      1. unpow2 43.5%

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

      2. sub-neg 43.5%

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

      3. unpow2 43.5%

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

      4. neg-mul-1 43.5%

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

      5. remove-double-neg 43.5%

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

      6. distribute-lft-out 43.5%

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

   9. Simplified 43.5%

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

   10. Taylor expanded in l around 0 56.1%

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

       1. associate-*l/ 56.2%

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

       2. *-lft-identity 56.2%

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

   12. Simplified 56.2%

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

   if 1.05000000000000005e-127 < t

   1. Initial program 40.1%

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

      1. associate-*l/ 40.2%

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

   3. Simplified 40.2%

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

   4. Taylor expanded in t around inf 32.3%

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

      1. +-commutative 32.3%

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

      2. associate-*r/ 48.5%

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

      3. sub-neg 48.5%

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

      4. metadata-eval 48.5%

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

      5. unpow2 48.5%

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

      6. +-commutative 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in x around inf 85.7%

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

      1. associate--l+ 85.7%

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

      2. associate-*r/ 85.7%

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

      3. metadata-eval 85.7%

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

      4. unpow2 85.7%

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

   9. Simplified 85.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-127}:\\ \;\;\;\;t \cdot \frac{\sqrt{x}}{\ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 9: 76.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-293}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -3.6e-293)
   (+ -1.0 (/ 1.0 x))
   (+ 1.0 (+ (/ 0.5 (* x x)) (/ -1.0 x)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-293) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.6d-293)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((0.5d0 / (x * x)) + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-293) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -3.6e-293:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -3.6e-293)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.6e-293)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + ((0.5 / (x * x)) + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -3.6e-293], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.59999999999999985e-293

   1. Initial program 31.9%

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

      1. associate-*l/ 31.9%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in t around inf 27.1%

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

      1. +-commutative 27.1%

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

      2. associate-*r/ 38.9%

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

      3. sub-neg 38.9%

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

      4. metadata-eval 38.9%

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

      5. unpow2 38.9%

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

      6. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in t around -inf 79.0%

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

      1. mul-1-neg 79.0%

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

      2. sub-neg 79.0%

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

      3. metadata-eval 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in x around inf 78.1%

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

   if -3.59999999999999985e-293 < t

   1. Initial program 32.1%

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

      1. associate-*l/ 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around inf 27.0%

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

      1. +-commutative 27.0%

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

      2. associate-*r/ 39.5%

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

      3. sub-neg 39.5%

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

      4. metadata-eval 39.5%

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

      5. unpow2 39.5%

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

      6. +-commutative 39.5%

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

   6. Simplified 39.5%

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

   7. Taylor expanded in x around inf 75.1%

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

      1. associate--l+ 75.1%

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

      2. associate-*r/ 75.1%

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

      3. metadata-eval 75.1%

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

      4. unpow2 75.1%

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

   9. Simplified 75.1%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-293}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 10: 77.2% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x} + \frac{-1}{x}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 - t_1\\ \mathbf{else}:\\ \;\;\;\;1 + t_1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ (/ 0.5 (* x x)) (/ -1.0 x))))
   (if (<= t -5e-310) (- -1.0 t_1) (+ 1.0 t_1))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (0.5 / (x * x)) + (-1.0 / x);
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 - t_1;
	} else {
		tmp = 1.0 + 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 = (0.5d0 / (x * x)) + ((-1.0d0) / x)
    if (t <= (-5d-310)) then
        tmp = (-1.0d0) - t_1
    else
        tmp = 1.0d0 + 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 = (0.5 / (x * x)) + (-1.0 / x);
	double tmp;
	if (t <= -5e-310) {
		tmp = -1.0 - t_1;
	} else {
		tmp = 1.0 + t_1;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (0.5 / (x * x)) + (-1.0 / x)
	tmp = 0
	if t <= -5e-310:
		tmp = -1.0 - t_1
	else:
		tmp = 1.0 + t_1
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(0.5 / Float64(x * x)) + Float64(-1.0 / x))
	tmp = 0.0
	if (t <= -5e-310)
		tmp = Float64(-1.0 - t_1);
	else
		tmp = Float64(1.0 + t_1);
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (0.5 / (x * x)) + (-1.0 / x);
	tmp = 0.0;
	if (t <= -5e-310)
		tmp = -1.0 - t_1;
	else
		tmp = 1.0 + 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[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5e-310], N[(-1.0 - t$95$1), $MachinePrecision], N[(1.0 + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.999999999999985e-310

   1. Initial program 31.6%

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

      1. associate-*l/ 31.7%

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

   3. Simplified 31.7%

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

   4. Taylor expanded in t around inf 26.9%

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

      1. +-commutative 26.9%

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

      2. associate-*r/ 38.6%

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

      3. sub-neg 38.6%

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

      4. metadata-eval 38.6%

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

      5. unpow2 38.6%

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

      6. +-commutative 38.6%

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

   6. Simplified 38.6%

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

   7. Taylor expanded in t around -inf 78.4%

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

      1. mul-1-neg 78.4%

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

      2. sub-neg 78.4%

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

      3. metadata-eval 78.4%

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

   9. Simplified 78.4%

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

   10. Taylor expanded in x around inf 77.7%

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

       1. associate--l+ 77.7%

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

       2. sub-neg 77.7%

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

       3. associate-*r/ 77.7%

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

       4. metadata-eval 77.7%

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

       5. unpow2 77.7%

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

       6. distribute-neg-frac 77.7%

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

       7. metadata-eval 77.7%

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

   12. Simplified 77.7%

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

   if -4.999999999999985e-310 < t

   1. Initial program 32.4%

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

      1. associate-*l/ 32.4%

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

   3. Simplified 32.4%

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

   4. Taylor expanded in t around inf 27.2%

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

      1. +-commutative 27.2%

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

      2. associate-*r/ 39.8%

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

      3. sub-neg 39.8%

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

      4. metadata-eval 39.8%

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

      5. unpow2 39.8%

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

      6. +-commutative 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in x around inf 75.6%

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

      1. associate--l+ 75.6%

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

      2. associate-*r/ 75.6%

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

      3. metadata-eval 75.6%

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

      4. unpow2 75.6%

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

   9. Simplified 75.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 - \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 11: 77.0% accurate, 17.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{0.5}{x \cdot x}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{x} - \left(1 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 + \frac{-1}{x}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ 0.5 (* x x))))
   (if (<= t -3.6e-293) (- (/ 1.0 x) (+ 1.0 t_1)) (+ 1.0 (+ t_1 (/ -1.0 x))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -3.6e-293) {
		tmp = (1.0 / x) - (1.0 + t_1);
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 0.5d0 / (x * x)
    if (t <= (-3.6d-293)) then
        tmp = (1.0d0 / x) - (1.0d0 + t_1)
    else
        tmp = 1.0d0 + (t_1 + ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = 0.5 / (x * x);
	double tmp;
	if (t <= -3.6e-293) {
		tmp = (1.0 / x) - (1.0 + t_1);
	} else {
		tmp = 1.0 + (t_1 + (-1.0 / x));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = 0.5 / (x * x)
	tmp = 0
	if t <= -3.6e-293:
		tmp = (1.0 / x) - (1.0 + t_1)
	else:
		tmp = 1.0 + (t_1 + (-1.0 / x))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(0.5 / Float64(x * x))
	tmp = 0.0
	if (t <= -3.6e-293)
		tmp = Float64(Float64(1.0 / x) - Float64(1.0 + t_1));
	else
		tmp = Float64(1.0 + Float64(t_1 + Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = 0.5 / (x * x);
	tmp = 0.0;
	if (t <= -3.6e-293)
		tmp = (1.0 / x) - (1.0 + t_1);
	else
		tmp = 1.0 + (t_1 + (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.6e-293], N[(N[(1.0 / x), $MachinePrecision] - N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.59999999999999985e-293

   1. Initial program 31.9%

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

      1. associate-*l/ 31.9%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in t around inf 27.1%

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

      1. +-commutative 27.1%

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

      2. associate-*r/ 38.9%

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

      3. sub-neg 38.9%

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

      4. metadata-eval 38.9%

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

      5. unpow2 38.9%

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

      6. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in t around -inf 79.0%

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

      1. mul-1-neg 79.0%

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

      2. sub-neg 79.0%

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

      3. metadata-eval 79.0%

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

   9. Simplified 79.0%

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

   10. Taylor expanded in x around inf 78.3%

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

       1. associate-*r/ 78.3%

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

       2. metadata-eval 78.3%

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

       3. unpow2 78.3%

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

   12. Simplified 78.3%

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

   if -3.59999999999999985e-293 < t

   1. Initial program 32.1%

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

      1. associate-*l/ 32.1%

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

   3. Simplified 32.1%

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

   4. Taylor expanded in t around inf 27.0%

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

      1. +-commutative 27.0%

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

      2. associate-*r/ 39.5%

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

      3. sub-neg 39.5%

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

      4. metadata-eval 39.5%

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

      5. unpow2 39.5%

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

      6. +-commutative 39.5%

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

   6. Simplified 39.5%

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

   7. Taylor expanded in x around inf 75.1%

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

      1. associate--l+ 75.1%

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

      2. associate-*r/ 75.1%

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

      3. metadata-eval 75.1%

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

      4. unpow2 75.1%

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

   9. Simplified 75.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-293}:\\ \;\;\;\;\frac{1}{x} - \left(1 + \frac{0.5}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{x \cdot x} + \frac{-1}{x}\right)\\ \end{array} \]

Alternative 12: 76.8% accurate, 31.9× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-293}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -3.6e-293) (+ -1.0 (/ 1.0 x)) (+ 1.0 (/ -1.0 x))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-293) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.6d-293)) then
        tmp = (-1.0d0) + (1.0d0 / x)
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -3.6e-293) {
		tmp = -1.0 + (1.0 / x);
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -3.6e-293:
		tmp = -1.0 + (1.0 / x)
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -3.6e-293)
		tmp = Float64(-1.0 + Float64(1.0 / x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -3.6e-293)
		tmp = -1.0 + (1.0 / x);
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -3.6e-293], N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.59999999999999985e-293

   1. Initial program 31.9%

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

      1. associate-*l/ 31.9%

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

   3. Simplified 31.9%

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

   4. Taylor expanded in t around inf 27.1%

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

      1. +-commutative 27.1%

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

      2. associate-*r/ 38.9%

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

      3. sub-neg 38.9%

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

      4. metadata-eval 38.9%

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

      5. unpow2 38.9%

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

      6. +-commutative 38.9%

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

   6. Simplified 38.9%

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

   7. Taylor expanded in t around -inf 79.0%

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

      1. mul-1-neg 79.0%

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

      2. sub-neg 79.0%

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

      3. metadata-eval 79.0%

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

   9. Simplified 79.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{x + -1}{1 + x}}} \]

   10. Taylor expanded in x around inf 78.1%

       \[\leadsto -\color{blue}{\left(1 - \frac{1}{x}\right)} \]

   if -3.59999999999999985e-293 < t

   1. Initial program 32.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Step-by-step derivation

      1. associate-*l/ 32.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

   3. Simplified 32.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

   4. Taylor expanded in t around inf 27.0%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}}}} \cdot t \]
   5. Step-by-step derivation

      1. +-commutative 27.0%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{{t}^{2} \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \cdot t \]

      2. associate-*r/ 39.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{x + 1}{x - 1}\right)}}} \cdot t \]

      3. sub-neg 39.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{\color{blue}{x + \left(-1\right)}}\right)}} \cdot t \]

      4. metadata-eval 39.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{x + \color{blue}{-1}}\right)}} \cdot t \]

      5. unpow2 39.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{x + 1}{x + -1}\right)}} \cdot t \]

      6. +-commutative 39.5%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \frac{x + 1}{\color{blue}{-1 + x}}\right)}} \cdot t \]

   6. Simplified 39.5%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\left(t \cdot t\right) \cdot \frac{x + 1}{-1 + x}\right)}}} \cdot t \]

   7. Taylor expanded in x around inf 75.0%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-293}:\\ \;\;\;\;-1 + \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 13: 39.2% accurate, 45.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ 1 + \frac{-1}{x} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (+ 1.0 (/ -1.0 x)))

C

l = abs(l);
double code(double x, double l, double t) {
	return 1.0 + (-1.0 / x);
}

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 + ((-1.0d0) / x)
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return 1.0 + (-1.0 / x);
}

Python

l = abs(l)
def code(x, l, t):
	return 1.0 + (-1.0 / x)

Julia

l = abs(l)
function code(x, l, t)
	return Float64(1.0 + Float64(-1.0 / x))
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = 1.0 + (-1.0 / x);
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 32.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 32.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around inf 27.1%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}}}} \cdot t \]
5. Step-by-step derivation

   1. +-commutative 27.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{{t}^{2} \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \cdot t \]

   2. associate-*r/ 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{x + 1}{x - 1}\right)}}} \cdot t \]

   3. sub-neg 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{\color{blue}{x + \left(-1\right)}}\right)}} \cdot t \]

   4. metadata-eval 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{x + \color{blue}{-1}}\right)}} \cdot t \]

   5. unpow2 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{x + 1}{x + -1}\right)}} \cdot t \]

   6. +-commutative 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \frac{x + 1}{\color{blue}{-1 + x}}\right)}} \cdot t \]

6. Simplified 39.2%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\left(t \cdot t\right) \cdot \frac{x + 1}{-1 + x}\right)}}} \cdot t \]

7. Taylor expanded in x around inf 39.2%

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]

8. Final simplification 39.2%

   \[\leadsto 1 + \frac{-1}{x} \]

Alternative 14: 38.9% 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 32.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Step-by-step derivation

   1. associate-*l/ 32.0%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \cdot t} \]

3. Simplified 32.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}} \cdot t} \]

4. Taylor expanded in t around inf 27.1%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \frac{{t}^{2} \cdot \left(1 + x\right)}{x - 1}}}} \cdot t \]
5. Step-by-step derivation

   1. +-commutative 27.1%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \frac{{t}^{2} \cdot \color{blue}{\left(x + 1\right)}}{x - 1}}} \cdot t \]

   2. associate-*r/ 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \color{blue}{\left({t}^{2} \cdot \frac{x + 1}{x - 1}\right)}}} \cdot t \]

   3. sub-neg 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{\color{blue}{x + \left(-1\right)}}\right)}} \cdot t \]

   4. metadata-eval 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left({t}^{2} \cdot \frac{x + 1}{x + \color{blue}{-1}}\right)}} \cdot t \]

   5. unpow2 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot \frac{x + 1}{x + -1}\right)}} \cdot t \]

   6. +-commutative 39.2%

      \[\leadsto \frac{\sqrt{2}}{\sqrt{2 \cdot \left(\left(t \cdot t\right) \cdot \frac{x + 1}{\color{blue}{-1 + x}}\right)}} \cdot t \]

6. Simplified 39.2%

   \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{2 \cdot \left(\left(t \cdot t\right) \cdot \frac{x + 1}{-1 + x}\right)}}} \cdot t \]

7. Taylor expanded in x around inf 39.2%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 39.2%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023293 
(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)))))