Toniolo and Linder, Equation (7)

Percentage Accurate: 32.7% → 81.7%

Time: 35.2s

Alternatives: 18

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: 32.7% 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: 81.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + 1}{-1 + x}}\\ t_3 := \frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)\\ t_4 := \frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{t_3}} - t \cdot \sqrt{t_3}}{t}}\\ t_5 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{t_2}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-300}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(0.5, \frac{2 \cdot t_5}{t \cdot \left(x \cdot \sqrt{2}\right)}, t_1\right)}{t}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(-t_5\right) - t_5}{x \cdot x}, \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right)\right) + \frac{t_5}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
        (t_3 (+ (/ 4.0 x) (+ 2.0 (/ 4.0 (* x x)))))
        (t_4
         (/
          (sqrt 2.0)
          (/
           (-
            (*
             (* -0.5 (/ (* (+ (/ 2.0 (* x x)) (/ 2.0 x)) (* l l)) t))
             (sqrt (/ 1.0 t_3)))
            (* t (sqrt t_3)))
           t)))
        (t_5 (fma 2.0 (* t t) (* l l))))
   (if (<= t -4.3e-32)
     (/ -1.0 t_2)
     (if (<= t -3.2e-186)
       t_4
       (if (<= t -1.75e-232)
         (* (/ (sqrt 2.0) l) (* t (pow (/ 2.0 x) -0.5)))
         (if (<= t -1.45e-300)
           t_4
           (if (<= t 5.8e-199)
             (/ (/ t l) (pow x -0.5))
             (if (<= t 6e-159)
               (/
                (sqrt 2.0)
                (/ (fma 0.5 (/ (* 2.0 t_5) (* t (* x (sqrt 2.0)))) t_1) t))
               (if (<= t 4.1e+58)
                 (/
                  (sqrt 2.0)
                  (/
                   (sqrt
                    (+
                     (fma
                      -1.0
                      (/ (- (- t_5) t_5) (* x x))
                      (fma 2.0 (/ (* t t) x) (fma 2.0 (* t t) (/ (* l l) x))))
                     (/ t_5 x)))
                   t))
                 (* t (/ (sqrt 2.0) (* t_1 t_2))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = sqrt(((x + 1.0) / (-1.0 + x)));
	double t_3 = (4.0 / x) + (2.0 + (4.0 / (x * x)));
	double t_4 = sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * sqrt((1.0 / t_3))) - (t * sqrt(t_3))) / t);
	double t_5 = fma(2.0, (t * t), (l * l));
	double tmp;
	if (t <= -4.3e-32) {
		tmp = -1.0 / t_2;
	} else if (t <= -3.2e-186) {
		tmp = t_4;
	} else if (t <= -1.75e-232) {
		tmp = (sqrt(2.0) / l) * (t * pow((2.0 / x), -0.5));
	} else if (t <= -1.45e-300) {
		tmp = t_4;
	} else if (t <= 5.8e-199) {
		tmp = (t / l) / pow(x, -0.5);
	} else if (t <= 6e-159) {
		tmp = sqrt(2.0) / (fma(0.5, ((2.0 * t_5) / (t * (x * sqrt(2.0)))), t_1) / t);
	} else if (t <= 4.1e+58) {
		tmp = sqrt(2.0) / (sqrt((fma(-1.0, ((-t_5 - t_5) / (x * x)), fma(2.0, ((t * t) / x), fma(2.0, (t * t), ((l * l) / x)))) + (t_5 / x))) / t);
	} else {
		tmp = t * (sqrt(2.0) / (t_1 * t_2));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	t_3 = Float64(Float64(4.0 / x) + Float64(2.0 + Float64(4.0 / Float64(x * x))))
	t_4 = Float64(sqrt(2.0) / Float64(Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)) * Float64(l * l)) / t)) * sqrt(Float64(1.0 / t_3))) - Float64(t * sqrt(t_3))) / t))
	t_5 = fma(2.0, Float64(t * t), Float64(l * l))
	tmp = 0.0
	if (t <= -4.3e-32)
		tmp = Float64(-1.0 / t_2);
	elseif (t <= -3.2e-186)
		tmp = t_4;
	elseif (t <= -1.75e-232)
		tmp = Float64(Float64(sqrt(2.0) / l) * Float64(t * (Float64(2.0 / x) ^ -0.5)));
	elseif (t <= -1.45e-300)
		tmp = t_4;
	elseif (t <= 5.8e-199)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	elseif (t <= 6e-159)
		tmp = Float64(sqrt(2.0) / Float64(fma(0.5, Float64(Float64(2.0 * t_5) / Float64(t * Float64(x * sqrt(2.0)))), t_1) / t));
	elseif (t <= 4.1e+58)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(fma(-1.0, Float64(Float64(Float64(-t_5) - t_5) / Float64(x * x)), fma(2.0, Float64(Float64(t * t) / x), fma(2.0, Float64(t * t), Float64(Float64(l * l) / x)))) + Float64(t_5 / x))) / t));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * 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[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 / x), $MachinePrecision] + N[(2.0 + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[(-0.5 * N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.3e-32], N[(-1.0 / t$95$2), $MachinePrecision], If[LessEqual[t, -3.2e-186], t$95$4, If[LessEqual[t, -1.75e-232], N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.45e-300], t$95$4, If[LessEqual[t, 5.8e-199], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-159], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(2.0 * t$95$5), $MachinePrecision] / N[(t * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.1e+58], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(-1.0 * N[(N[((-t$95$5) - t$95$5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -4.2999999999999999e-32

   1. Initial program 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in t around -inf 85.8%

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

      1. associate-*r* 85.8%

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

      2. neg-mul-1 85.8%

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

      3. +-commutative 85.8%

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

      4. sub-neg 85.8%

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

      5. metadata-eval 85.8%

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

      6. +-commutative 85.8%

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

   6. Simplified 85.8%

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

      1. frac-2neg 85.8%

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

      2. neg-mul-1 85.8%

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

      3. *-commutative 85.8%

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

      4. distribute-lft-neg-in 85.8%

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

      5. times-frac 85.8%

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

      6. neg-mul-1 85.8%

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

      7. *-commutative 85.8%

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

      8. distribute-rgt-neg-in 85.8%

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

      9. metadata-eval 85.8%

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

      10. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

      3. *-inverses 85.8%

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

      4. metadata-eval 85.8%

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

   10. Simplified 85.8%

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

   if -4.2999999999999999e-32 < t < -3.2e-186 or -1.7499999999999999e-232 < t < -1.44999999999999996e-300

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in x around -inf 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around -inf 78.9%

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

   9. Simplified 81.8%

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

   if -3.2e-186 < t < -1.7499999999999999e-232

   1. Initial program 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in l around inf 1.0%

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

   5. Taylor expanded in x around inf 24.1%

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

      1. associate-/r* 24.1%

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

      2. div-inv 24.1%

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

      3. associate-/r/ 24.1%

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

      4. pow1/2 24.1%

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

      5. pow-flip 24.1%

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

      6. metadata-eval 24.1%

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

   7. Applied egg-rr 24.1%

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

      1. associate-*l* 39.8%

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

   9. Simplified 39.8%

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

   if -1.44999999999999996e-300 < t < 5.8e-199

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in l around inf 0.7%

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

   5. Taylor expanded in x around inf 46.8%

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

      1. associate-/r* 46.8%

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

      2. div-inv 46.8%

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

      3. times-frac 46.8%

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

      4. *-commutative 46.8%

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

      5. times-frac 46.8%

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

      6. clear-num 46.8%

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

      7. times-frac 46.7%

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

      8. *-commutative 46.7%

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

      9. sqrt-div 46.7%

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

      10. associate-*r/ 46.8%

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

      11. un-div-inv 46.8%

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

      12. metadata-eval 46.8%

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

      13. sqrt-div 46.8%

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

      14. *-commutative 46.8%

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

      15. times-frac 46.8%

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

      16. inv-pow 46.8%

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

      17. sqrt-pow1 46.8%

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

      18. metadata-eval 46.8%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 46.8%

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

      1. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 46.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 46.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 46.9%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 46.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 46.9%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 46.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 46.9%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 5.8e-199 < t < 6.00000000000000018e-159

   1. Initial program 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in x around -inf 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in x around inf 89.3%

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

      1. fma-def 89.3%

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

   9. Simplified 89.3%

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

   if 6.00000000000000018e-159 < t < 4.1e58

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in x around -inf 78.2%

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

   6. Simplified 78.2%

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

   if 4.1e58 < t

   1. Initial program 38.1%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in t around inf 93.6%

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

      1. +-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

   6. Simplified 93.6%

      \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-159}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}{t}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+58}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(-1, \frac{\left(-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\right) - \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x \cdot x}, \mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 2: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{\frac{x + 1}{-1 + x}}\\ t_3 := \frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)\\ t_4 := \frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{t_3}} - t \cdot \sqrt{t_3}}{t}}\\ t_5 := \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{t_2}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-186}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(0.5, \frac{2 \cdot t_5}{t \cdot \left(x \cdot \sqrt{2}\right)}, t_1\right)}{t}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{t_5}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_1 \cdot t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0)))
        (t_2 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
        (t_3 (+ (/ 4.0 x) (+ 2.0 (/ 4.0 (* x x)))))
        (t_4
         (/
          (sqrt 2.0)
          (/
           (-
            (*
             (* -0.5 (/ (* (+ (/ 2.0 (* x x)) (/ 2.0 x)) (* l l)) t))
             (sqrt (/ 1.0 t_3)))
            (* t (sqrt t_3)))
           t)))
        (t_5 (fma 2.0 (* t t) (* l l))))
   (if (<= t -4.8e-34)
     (/ -1.0 t_2)
     (if (<= t -4.8e-186)
       t_4
       (if (<= t -7.5e-233)
         (* (/ (sqrt 2.0) l) (* t (pow (/ 2.0 x) -0.5)))
         (if (<= t -4.6e-299)
           t_4
           (if (<= t 5.6e-199)
             (/ (/ t l) (pow x -0.5))
             (if (<= t 8.5e-160)
               (/
                (sqrt 2.0)
                (/ (fma 0.5 (/ (* 2.0 t_5) (* t (* x (sqrt 2.0)))) t_1) t))
               (if (<= t 5.2e+59)
                 (/
                  (sqrt 2.0)
                  (/
                   (sqrt
                    (+
                     (fma 2.0 (/ (* t t) x) (fma 2.0 (* t t) (/ (* l l) x)))
                     (/ t_5 x)))
                   t))
                 (* t (/ (sqrt 2.0) (* t_1 t_2))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = t * sqrt(2.0);
	double t_2 = sqrt(((x + 1.0) / (-1.0 + x)));
	double t_3 = (4.0 / x) + (2.0 + (4.0 / (x * x)));
	double t_4 = sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * sqrt((1.0 / t_3))) - (t * sqrt(t_3))) / t);
	double t_5 = fma(2.0, (t * t), (l * l));
	double tmp;
	if (t <= -4.8e-34) {
		tmp = -1.0 / t_2;
	} else if (t <= -4.8e-186) {
		tmp = t_4;
	} else if (t <= -7.5e-233) {
		tmp = (sqrt(2.0) / l) * (t * pow((2.0 / x), -0.5));
	} else if (t <= -4.6e-299) {
		tmp = t_4;
	} else if (t <= 5.6e-199) {
		tmp = (t / l) / pow(x, -0.5);
	} else if (t <= 8.5e-160) {
		tmp = sqrt(2.0) / (fma(0.5, ((2.0 * t_5) / (t * (x * sqrt(2.0)))), t_1) / t);
	} else if (t <= 5.2e+59) {
		tmp = sqrt(2.0) / (sqrt((fma(2.0, ((t * t) / x), fma(2.0, (t * t), ((l * l) / x))) + (t_5 / x))) / t);
	} else {
		tmp = t * (sqrt(2.0) / (t_1 * t_2));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(t * sqrt(2.0))
	t_2 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	t_3 = Float64(Float64(4.0 / x) + Float64(2.0 + Float64(4.0 / Float64(x * x))))
	t_4 = Float64(sqrt(2.0) / Float64(Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)) * Float64(l * l)) / t)) * sqrt(Float64(1.0 / t_3))) - Float64(t * sqrt(t_3))) / t))
	t_5 = fma(2.0, Float64(t * t), Float64(l * l))
	tmp = 0.0
	if (t <= -4.8e-34)
		tmp = Float64(-1.0 / t_2);
	elseif (t <= -4.8e-186)
		tmp = t_4;
	elseif (t <= -7.5e-233)
		tmp = Float64(Float64(sqrt(2.0) / l) * Float64(t * (Float64(2.0 / x) ^ -0.5)));
	elseif (t <= -4.6e-299)
		tmp = t_4;
	elseif (t <= 5.6e-199)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	elseif (t <= 8.5e-160)
		tmp = Float64(sqrt(2.0) / Float64(fma(0.5, Float64(Float64(2.0 * t_5) / Float64(t * Float64(x * sqrt(2.0)))), t_1) / t));
	elseif (t <= 5.2e+59)
		tmp = Float64(sqrt(2.0) / Float64(sqrt(Float64(fma(2.0, Float64(Float64(t * t) / x), fma(2.0, Float64(t * t), Float64(Float64(l * l) / x))) + Float64(t_5 / x))) / t));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_1 * 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[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 / x), $MachinePrecision] + N[(2.0 + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[(-0.5 * N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e-34], N[(-1.0 / t$95$2), $MachinePrecision], If[LessEqual[t, -4.8e-186], t$95$4, If[LessEqual[t, -7.5e-233], N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.6e-299], t$95$4, If[LessEqual[t, 5.6e-199], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-160], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(2.0 * t$95$5), $MachinePrecision] / N[(t * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.2e+59], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[Sqrt[N[(N[(2.0 * N[(N[(t * t), $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision] + N[(N[(l * l), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -4.79999999999999982e-34

   1. Initial program 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in t around -inf 85.8%

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

      1. associate-*r* 85.8%

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

      2. neg-mul-1 85.8%

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

      3. +-commutative 85.8%

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

      4. sub-neg 85.8%

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

      5. metadata-eval 85.8%

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

      6. +-commutative 85.8%

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

   6. Simplified 85.8%

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

      1. frac-2neg 85.8%

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

      2. neg-mul-1 85.8%

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

      3. *-commutative 85.8%

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

      4. distribute-lft-neg-in 85.8%

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

      5. times-frac 85.8%

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

      6. neg-mul-1 85.8%

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

      7. *-commutative 85.8%

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

      8. distribute-rgt-neg-in 85.8%

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

      9. metadata-eval 85.8%

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

      10. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

      3. *-inverses 85.8%

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

      4. metadata-eval 85.8%

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

   10. Simplified 85.8%

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

   if -4.79999999999999982e-34 < t < -4.80000000000000006e-186 or -7.49999999999999974e-233 < t < -4.6000000000000001e-299

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in x around -inf 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around -inf 78.9%

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

   9. Simplified 81.8%

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

   if -4.80000000000000006e-186 < t < -7.49999999999999974e-233

   1. Initial program 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in l around inf 1.0%

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

   5. Taylor expanded in x around inf 24.1%

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

      1. associate-/r* 24.1%

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

      2. div-inv 24.1%

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

      3. associate-/r/ 24.1%

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

      4. pow1/2 24.1%

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

      5. pow-flip 24.1%

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

      6. metadata-eval 24.1%

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

   7. Applied egg-rr 24.1%

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

      1. associate-*l* 39.8%

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

   9. Simplified 39.8%

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

   if -4.6000000000000001e-299 < t < 5.60000000000000036e-199

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in l around inf 0.7%

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

   5. Taylor expanded in x around inf 46.8%

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

      1. associate-/r* 46.8%

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

      2. div-inv 46.8%

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

      3. times-frac 46.8%

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

      4. *-commutative 46.8%

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

      5. times-frac 46.8%

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

      6. clear-num 46.8%

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

      7. times-frac 46.7%

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

      8. *-commutative 46.7%

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

      9. sqrt-div 46.7%

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

      10. associate-*r/ 46.8%

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

      11. un-div-inv 46.8%

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

      12. metadata-eval 46.8%

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

      13. sqrt-div 46.8%

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

      14. *-commutative 46.8%

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

      15. times-frac 46.8%

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

      16. inv-pow 46.8%

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

      17. sqrt-pow1 46.8%

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

      18. metadata-eval 46.8%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 46.8%

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

      1. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 46.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 46.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 46.9%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 46.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 46.9%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 46.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 46.9%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 5.60000000000000036e-199 < t < 8.49999999999999959e-160

   1. Initial program 12.4%

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

   3. Simplified 12.4%

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

   4. Taylor expanded in x around -inf 23.1%

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

   6. Simplified 23.1%

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

   7. Taylor expanded in x around inf 89.3%

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

      1. fma-def 89.3%

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

   9. Simplified 89.3%

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

   if 8.49999999999999959e-160 < t < 5.19999999999999999e59

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in x around inf 77.9%

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

      1. fma-def 77.9%

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

      2. unpow2 77.9%

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

      3. fma-def 77.9%

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

      4. unpow2 77.9%

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

      5. unpow2 77.9%

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

      6. associate-*r/ 77.9%

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

      7. mul-1-neg 77.9%

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

      8. unpow2 77.9%

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

      9. fma-def 77.9%

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

      10. unpow2 77.9%

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

   6. Simplified 77.9%

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

   if 5.19999999999999999e59 < t

   1. Initial program 38.1%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in t around inf 93.6%

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

      1. +-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

   6. Simplified 93.6%

      \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}{t}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) + \frac{\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 3: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)\\ t_2 := \frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{t_1}} - t \cdot \sqrt{t_1}}{t}}\\ t_3 := t \cdot \sqrt{2}\\ t_4 := \sqrt{\frac{x + 1}{-1 + x}}\\ \mathbf{if}\;t \leq -1.12 \cdot 10^{-31}:\\ \;\;\;\;\frac{-1}{t_4}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-299}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t_3\right)}{t}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-70}:\\ \;\;\;\;\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{t_3 \cdot t_4}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ (/ 4.0 x) (+ 2.0 (/ 4.0 (* x x)))))
        (t_2
         (/
          (sqrt 2.0)
          (/
           (-
            (*
             (* -0.5 (/ (* (+ (/ 2.0 (* x x)) (/ 2.0 x)) (* l l)) t))
             (sqrt (/ 1.0 t_1)))
            (* t (sqrt t_1)))
           t)))
        (t_3 (* t (sqrt 2.0)))
        (t_4 (sqrt (/ (+ x 1.0) (+ -1.0 x)))))
   (if (<= t -1.12e-31)
     (/ -1.0 t_4)
     (if (<= t -7.5e-186)
       t_2
       (if (<= t -8e-233)
         (* (/ (sqrt 2.0) l) (* t (pow (/ 2.0 x) -0.5)))
         (if (<= t -4.8e-299)
           t_2
           (if (<= t 5.6e-199)
             (/ (/ t l) (pow x -0.5))
             (if (<= t 1.02e-122)
               (/
                (sqrt 2.0)
                (/
                 (fma
                  0.5
                  (/ (* 2.0 (fma 2.0 (* t t) (* l l))) (* t (* x (sqrt 2.0))))
                  t_3)
                 t))
               (if (<= t 2.05e-70)
                 (*
                  (/ (pow 2.0 0.25) (/ 1.0 t))
                  (/ (pow 2.0 0.25) (* l (sqrt (/ 2.0 x)))))
                 (* t (/ (sqrt 2.0) (* t_3 t_4))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (4.0 / x) + (2.0 + (4.0 / (x * x)));
	double t_2 = sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * sqrt((1.0 / t_1))) - (t * sqrt(t_1))) / t);
	double t_3 = t * sqrt(2.0);
	double t_4 = sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -1.12e-31) {
		tmp = -1.0 / t_4;
	} else if (t <= -7.5e-186) {
		tmp = t_2;
	} else if (t <= -8e-233) {
		tmp = (sqrt(2.0) / l) * (t * pow((2.0 / x), -0.5));
	} else if (t <= -4.8e-299) {
		tmp = t_2;
	} else if (t <= 5.6e-199) {
		tmp = (t / l) / pow(x, -0.5);
	} else if (t <= 1.02e-122) {
		tmp = sqrt(2.0) / (fma(0.5, ((2.0 * fma(2.0, (t * t), (l * l))) / (t * (x * sqrt(2.0)))), t_3) / t);
	} else if (t <= 2.05e-70) {
		tmp = (pow(2.0, 0.25) / (1.0 / t)) * (pow(2.0, 0.25) / (l * sqrt((2.0 / x))));
	} else {
		tmp = t * (sqrt(2.0) / (t_3 * t_4));
	}
	return tmp;
}

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(4.0 / x) + Float64(2.0 + Float64(4.0 / Float64(x * x))))
	t_2 = Float64(sqrt(2.0) / Float64(Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)) * Float64(l * l)) / t)) * sqrt(Float64(1.0 / t_1))) - Float64(t * sqrt(t_1))) / t))
	t_3 = Float64(t * sqrt(2.0))
	t_4 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	tmp = 0.0
	if (t <= -1.12e-31)
		tmp = Float64(-1.0 / t_4);
	elseif (t <= -7.5e-186)
		tmp = t_2;
	elseif (t <= -8e-233)
		tmp = Float64(Float64(sqrt(2.0) / l) * Float64(t * (Float64(2.0 / x) ^ -0.5)));
	elseif (t <= -4.8e-299)
		tmp = t_2;
	elseif (t <= 5.6e-199)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	elseif (t <= 1.02e-122)
		tmp = Float64(sqrt(2.0) / Float64(fma(0.5, Float64(Float64(2.0 * fma(2.0, Float64(t * t), Float64(l * l))) / Float64(t * Float64(x * sqrt(2.0)))), t_3) / t));
	elseif (t <= 2.05e-70)
		tmp = Float64(Float64((2.0 ^ 0.25) / Float64(1.0 / t)) * Float64((2.0 ^ 0.25) / Float64(l * sqrt(Float64(2.0 / x)))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(t_3 * t_4)));
	end
	return tmp
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(4.0 / x), $MachinePrecision] + N[(2.0 + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[(-0.5 * N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.12e-31], N[(-1.0 / t$95$4), $MachinePrecision], If[LessEqual[t, -7.5e-186], t$95$2, If[LessEqual[t, -8e-233], N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.8e-299], t$95$2, If[LessEqual[t, 5.6e-199], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e-122], N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(0.5 * N[(N[(2.0 * N[(2.0 * N[(t * t), $MachinePrecision] + N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e-70], N[(N[(N[Power[2.0, 0.25], $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.12e-31

   1. Initial program 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in t around -inf 85.8%

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

      1. associate-*r* 85.8%

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

      2. neg-mul-1 85.8%

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

      3. +-commutative 85.8%

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

      4. sub-neg 85.8%

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

      5. metadata-eval 85.8%

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

      6. +-commutative 85.8%

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

   6. Simplified 85.8%

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

      1. frac-2neg 85.8%

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

      2. neg-mul-1 85.8%

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

      3. *-commutative 85.8%

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

      4. distribute-lft-neg-in 85.8%

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

      5. times-frac 85.8%

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

      6. neg-mul-1 85.8%

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

      7. *-commutative 85.8%

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

      8. distribute-rgt-neg-in 85.8%

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

      9. metadata-eval 85.8%

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

      10. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

      3. *-inverses 85.8%

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

      4. metadata-eval 85.8%

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

   10. Simplified 85.8%

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

   if -1.12e-31 < t < -7.50000000000000076e-186 or -7.99999999999999966e-233 < t < -4.80000000000000039e-299

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in x around -inf 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around -inf 78.9%

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

   9. Simplified 81.8%

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

   if -7.50000000000000076e-186 < t < -7.99999999999999966e-233

   1. Initial program 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in l around inf 1.0%

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

   5. Taylor expanded in x around inf 24.1%

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

      1. associate-/r* 24.1%

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

      2. div-inv 24.1%

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

      3. associate-/r/ 24.1%

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

      4. pow1/2 24.1%

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

      5. pow-flip 24.1%

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

      6. metadata-eval 24.1%

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

   7. Applied egg-rr 24.1%

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

      1. associate-*l* 39.8%

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

   9. Simplified 39.8%

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

   if -4.80000000000000039e-299 < t < 5.60000000000000036e-199

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in l around inf 0.7%

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

   5. Taylor expanded in x around inf 46.8%

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

      1. associate-/r* 46.8%

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

      2. div-inv 46.8%

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

      3. times-frac 46.8%

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

      4. *-commutative 46.8%

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

      5. times-frac 46.8%

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

      6. clear-num 46.8%

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

      7. times-frac 46.7%

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

      8. *-commutative 46.7%

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

      9. sqrt-div 46.7%

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

      10. associate-*r/ 46.8%

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

      11. un-div-inv 46.8%

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

      12. metadata-eval 46.8%

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

      13. sqrt-div 46.8%

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

      14. *-commutative 46.8%

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

      15. times-frac 46.8%

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

      16. inv-pow 46.8%

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

      17. sqrt-pow1 46.8%

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

      18. metadata-eval 46.8%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 46.8%

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

      1. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 46.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 46.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 46.9%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 46.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 46.9%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 46.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 46.9%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 5.60000000000000036e-199 < t < 1.02000000000000002e-122

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around -inf 53.8%

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

   6. Simplified 53.8%

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

   7. Taylor expanded in x around inf 87.2%

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

      1. fma-def 87.2%

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

   9. Simplified 87.2%

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

   if 1.02000000000000002e-122 < t < 2.04999999999999989e-70

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in l around inf 1.7%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. pow1/2 38.8%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}}}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}} \]

      2. sqr-pow 38.6%

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

      3. *-commutative 38.6%

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

      4. div-inv 38.6%

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

      5. associate-*r* 38.6%

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

      6. *-commutative 38.6%

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

      7. sqrt-div 38.4%

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

      8. associate-*r/ 38.2%

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

      9. un-div-inv 38.2%

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

      10. metadata-eval 38.2%

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

      11. sqrt-div 38.5%

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

      12. *-commutative 38.5%

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

      13. times-frac 45.6%

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

      14. metadata-eval 45.6%

          \[\leadsto \frac{{2}^{\color{blue}{0.25}}}{\frac{1}{t}} \cdot \frac{{2}^{\left(\frac{0.5}{2}\right)}}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}} \]

      15. metadata-eval 45.6%

          \[\leadsto \frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}} \]

      16. sqrt-div 45.5%

          \[\leadsto \frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}} \]

   7. Applied egg-rr 45.8%

      \[\leadsto \color{blue}{\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}} \]

   if 2.04999999999999989e-70 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in t around inf 88.2%

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

      1. +-commutative 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

   6. Simplified 88.2%

      \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{-31}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\mathsf{fma}\left(0.5, \frac{2 \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{t \cdot \left(x \cdot \sqrt{2}\right)}, t \cdot \sqrt{2}\right)}{t}}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-70}:\\ \;\;\;\;\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 4: 78.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)\\ t_2 := \frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{t_1}} - t \cdot \sqrt{t_1}}{t}}\\ t_3 := \sqrt{\frac{x + 1}{-1 + x}}\\ \mathbf{if}\;t \leq -9 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{t_3}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-186}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-198}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot t_3}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (+ (/ 4.0 x) (+ 2.0 (/ 4.0 (* x x)))))
        (t_2
         (/
          (sqrt 2.0)
          (/
           (-
            (*
             (* -0.5 (/ (* (+ (/ 2.0 (* x x)) (/ 2.0 x)) (* l l)) t))
             (sqrt (/ 1.0 t_1)))
            (* t (sqrt t_1)))
           t)))
        (t_3 (sqrt (/ (+ x 1.0) (+ -1.0 x)))))
   (if (<= t -9e-32)
     (/ -1.0 t_3)
     (if (<= t -3.7e-186)
       t_2
       (if (<= t -1.75e-232)
         (* (/ (sqrt 2.0) l) (* t (pow (/ 2.0 x) -0.5)))
         (if (<= t -2.8e-301)
           t_2
           (if (<= t 1e-198)
             (/ (/ t l) (pow x -0.5))
             (if (<= t 1.02e-122)
               1.0
               (if (<= t 1.9e-69)
                 (*
                  (/ (pow 2.0 0.25) (/ 1.0 t))
                  (/ (pow 2.0 0.25) (* l (sqrt (/ 2.0 x)))))
                 (* t (/ (sqrt 2.0) (* (* t (sqrt 2.0)) t_3))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (4.0 / x) + (2.0 + (4.0 / (x * x)));
	double t_2 = sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * sqrt((1.0 / t_1))) - (t * sqrt(t_1))) / t);
	double t_3 = sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -9e-32) {
		tmp = -1.0 / t_3;
	} else if (t <= -3.7e-186) {
		tmp = t_2;
	} else if (t <= -1.75e-232) {
		tmp = (sqrt(2.0) / l) * (t * pow((2.0 / x), -0.5));
	} else if (t <= -2.8e-301) {
		tmp = t_2;
	} else if (t <= 1e-198) {
		tmp = (t / l) / pow(x, -0.5);
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 1.9e-69) {
		tmp = (pow(2.0, 0.25) / (1.0 / t)) * (pow(2.0, 0.25) / (l * sqrt((2.0 / x))));
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * t_3));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (4.0d0 / x) + (2.0d0 + (4.0d0 / (x * x)))
    t_2 = sqrt(2.0d0) / (((((-0.5d0) * ((((2.0d0 / (x * x)) + (2.0d0 / x)) * (l * l)) / t)) * sqrt((1.0d0 / t_1))) - (t * sqrt(t_1))) / t)
    t_3 = sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    if (t <= (-9d-32)) then
        tmp = (-1.0d0) / t_3
    else if (t <= (-3.7d-186)) then
        tmp = t_2
    else if (t <= (-1.75d-232)) then
        tmp = (sqrt(2.0d0) / l) * (t * ((2.0d0 / x) ** (-0.5d0)))
    else if (t <= (-2.8d-301)) then
        tmp = t_2
    else if (t <= 1d-198) then
        tmp = (t / l) / (x ** (-0.5d0))
    else if (t <= 1.02d-122) then
        tmp = 1.0d0
    else if (t <= 1.9d-69) then
        tmp = ((2.0d0 ** 0.25d0) / (1.0d0 / t)) * ((2.0d0 ** 0.25d0) / (l * sqrt((2.0d0 / x))))
    else
        tmp = t * (sqrt(2.0d0) / ((t * sqrt(2.0d0)) * t_3))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = (4.0 / x) + (2.0 + (4.0 / (x * x)));
	double t_2 = Math.sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * Math.sqrt((1.0 / t_1))) - (t * Math.sqrt(t_1))) / t);
	double t_3 = Math.sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -9e-32) {
		tmp = -1.0 / t_3;
	} else if (t <= -3.7e-186) {
		tmp = t_2;
	} else if (t <= -1.75e-232) {
		tmp = (Math.sqrt(2.0) / l) * (t * Math.pow((2.0 / x), -0.5));
	} else if (t <= -2.8e-301) {
		tmp = t_2;
	} else if (t <= 1e-198) {
		tmp = (t / l) / Math.pow(x, -0.5);
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 1.9e-69) {
		tmp = (Math.pow(2.0, 0.25) / (1.0 / t)) * (Math.pow(2.0, 0.25) / (l * Math.sqrt((2.0 / x))));
	} else {
		tmp = t * (Math.sqrt(2.0) / ((t * Math.sqrt(2.0)) * t_3));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (4.0 / x) + (2.0 + (4.0 / (x * x)))
	t_2 = math.sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * math.sqrt((1.0 / t_1))) - (t * math.sqrt(t_1))) / t)
	t_3 = math.sqrt(((x + 1.0) / (-1.0 + x)))
	tmp = 0
	if t <= -9e-32:
		tmp = -1.0 / t_3
	elif t <= -3.7e-186:
		tmp = t_2
	elif t <= -1.75e-232:
		tmp = (math.sqrt(2.0) / l) * (t * math.pow((2.0 / x), -0.5))
	elif t <= -2.8e-301:
		tmp = t_2
	elif t <= 1e-198:
		tmp = (t / l) / math.pow(x, -0.5)
	elif t <= 1.02e-122:
		tmp = 1.0
	elif t <= 1.9e-69:
		tmp = (math.pow(2.0, 0.25) / (1.0 / t)) * (math.pow(2.0, 0.25) / (l * math.sqrt((2.0 / x))))
	else:
		tmp = t * (math.sqrt(2.0) / ((t * math.sqrt(2.0)) * t_3))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(4.0 / x) + Float64(2.0 + Float64(4.0 / Float64(x * x))))
	t_2 = Float64(sqrt(2.0) / Float64(Float64(Float64(Float64(-0.5 * Float64(Float64(Float64(Float64(2.0 / Float64(x * x)) + Float64(2.0 / x)) * Float64(l * l)) / t)) * sqrt(Float64(1.0 / t_1))) - Float64(t * sqrt(t_1))) / t))
	t_3 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	tmp = 0.0
	if (t <= -9e-32)
		tmp = Float64(-1.0 / t_3);
	elseif (t <= -3.7e-186)
		tmp = t_2;
	elseif (t <= -1.75e-232)
		tmp = Float64(Float64(sqrt(2.0) / l) * Float64(t * (Float64(2.0 / x) ^ -0.5)));
	elseif (t <= -2.8e-301)
		tmp = t_2;
	elseif (t <= 1e-198)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 1.9e-69)
		tmp = Float64(Float64((2.0 ^ 0.25) / Float64(1.0 / t)) * Float64((2.0 ^ 0.25) / Float64(l * sqrt(Float64(2.0 / x)))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * t_3)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (4.0 / x) + (2.0 + (4.0 / (x * x)));
	t_2 = sqrt(2.0) / ((((-0.5 * ((((2.0 / (x * x)) + (2.0 / x)) * (l * l)) / t)) * sqrt((1.0 / t_1))) - (t * sqrt(t_1))) / t);
	t_3 = sqrt(((x + 1.0) / (-1.0 + x)));
	tmp = 0.0;
	if (t <= -9e-32)
		tmp = -1.0 / t_3;
	elseif (t <= -3.7e-186)
		tmp = t_2;
	elseif (t <= -1.75e-232)
		tmp = (sqrt(2.0) / l) * (t * ((2.0 / x) ^ -0.5));
	elseif (t <= -2.8e-301)
		tmp = t_2;
	elseif (t <= 1e-198)
		tmp = (t / l) / (x ^ -0.5);
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 1.9e-69)
		tmp = ((2.0 ^ 0.25) / (1.0 / t)) * ((2.0 ^ 0.25) / (l * sqrt((2.0 / x))));
	else
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * t_3));
	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[(4.0 / x), $MachinePrecision] + N[(2.0 + N[(4.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(N[(N[(-0.5 * N[(N[(N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -9e-32], N[(-1.0 / t$95$3), $MachinePrecision], If[LessEqual[t, -3.7e-186], t$95$2, If[LessEqual[t, -1.75e-232], N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.8e-301], t$95$2, If[LessEqual[t, 1e-198], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.02e-122], 1.0, If[LessEqual[t, 1.9e-69], N[(N[(N[Power[2.0, 0.25], $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -9.00000000000000009e-32

   1. Initial program 35.2%

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

   3. Simplified 35.2%

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

   4. Taylor expanded in t around -inf 85.8%

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

      1. associate-*r* 85.8%

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

      2. neg-mul-1 85.8%

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

      3. +-commutative 85.8%

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

      4. sub-neg 85.8%

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

      5. metadata-eval 85.8%

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

      6. +-commutative 85.8%

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

   6. Simplified 85.8%

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

      1. frac-2neg 85.8%

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

      2. neg-mul-1 85.8%

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

      3. *-commutative 85.8%

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

      4. distribute-lft-neg-in 85.8%

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

      5. times-frac 85.8%

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

      6. neg-mul-1 85.8%

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

      7. *-commutative 85.8%

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

      8. distribute-rgt-neg-in 85.8%

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

      9. metadata-eval 85.8%

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

      10. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.8%

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

      2. *-rgt-identity 85.8%

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

      3. *-inverses 85.8%

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

      4. metadata-eval 85.8%

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

   10. Simplified 85.8%

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

   if -9.00000000000000009e-32 < t < -3.7000000000000002e-186 or -1.7499999999999999e-232 < t < -2.8000000000000001e-301

   1. Initial program 24.2%

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

   3. Simplified 24.2%

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

   4. Taylor expanded in x around -inf 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in t around -inf 78.9%

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

   9. Simplified 81.8%

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

   if -3.7000000000000002e-186 < t < -1.7499999999999999e-232

   1. Initial program 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in l around inf 1.0%

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

   5. Taylor expanded in x around inf 24.1%

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

      1. associate-/r* 24.1%

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

      2. div-inv 24.1%

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

      3. associate-/r/ 24.1%

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

      4. pow1/2 24.1%

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

      5. pow-flip 24.1%

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

      6. metadata-eval 24.1%

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

   7. Applied egg-rr 24.1%

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

      1. associate-*l* 39.8%

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

   9. Simplified 39.8%

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

   if -2.8000000000000001e-301 < t < 9.9999999999999991e-199

   1. Initial program 2.0%

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

   3. Simplified 2.0%

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

   4. Taylor expanded in l around inf 0.7%

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

   5. Taylor expanded in x around inf 46.8%

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

      1. associate-/r* 46.8%

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

      2. div-inv 46.8%

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

      3. times-frac 46.8%

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

      4. *-commutative 46.8%

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

      5. times-frac 46.8%

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

      6. clear-num 46.8%

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

      7. times-frac 46.7%

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

      8. *-commutative 46.7%

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

      9. sqrt-div 46.7%

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

      10. associate-*r/ 46.8%

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

      11. un-div-inv 46.8%

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

      12. metadata-eval 46.8%

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

      13. sqrt-div 46.8%

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

      14. *-commutative 46.8%

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

      15. times-frac 46.8%

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

      16. inv-pow 46.8%

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

      17. sqrt-pow1 46.8%

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

      18. metadata-eval 46.8%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 46.8%

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

      1. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 46.9%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 46.9%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 46.9%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 46.9%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 46.9%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 46.9%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 46.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 46.9%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 46.9%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 9.9999999999999991e-199 < t < 1.02000000000000002e-122

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

      1. rewrite-binary64/binary32 48.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 48.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 86.7%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 86.7%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 86.7%

         \[\leadsto 1 \]

   10. Applied egg-rr 86.7%

       \[\leadsto 1 \]

   if 1.02000000000000002e-122 < t < 1.8999999999999999e-69

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in l around inf 1.7%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. pow1/2 38.8%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}}}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}} \]

      2. sqr-pow 38.6%

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

      3. *-commutative 38.6%

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

      4. div-inv 38.6%

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

      5. associate-*r* 38.6%

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

      6. *-commutative 38.6%

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

      7. sqrt-div 38.4%

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

      8. associate-*r/ 38.2%

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

      9. un-div-inv 38.2%

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

      10. metadata-eval 38.2%

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

      11. sqrt-div 38.5%

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

      12. *-commutative 38.5%

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

      13. times-frac 45.6%

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

      14. metadata-eval 45.6%

          \[\leadsto \frac{{2}^{\color{blue}{0.25}}}{\frac{1}{t}} \cdot \frac{{2}^{\left(\frac{0.5}{2}\right)}}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}} \]

      15. metadata-eval 45.6%

          \[\leadsto \frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}} \]

      16. sqrt-div 45.5%

          \[\leadsto \frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}} \]

   7. Applied egg-rr 45.8%

      \[\leadsto \color{blue}{\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}} \]

   if 1.8999999999999999e-69 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in t around inf 88.2%

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

      1. +-commutative 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

   6. Simplified 88.2%

      \[\leadsto t \cdot \frac{\sqrt{2}}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{2}}{\frac{\left(-0.5 \cdot \frac{\left(\frac{2}{x \cdot x} + \frac{2}{x}\right) \cdot \left(\ell \cdot \ell\right)}{t}\right) \cdot \sqrt{\frac{1}{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}} - t \cdot \sqrt{\frac{4}{x} + \left(2 + \frac{4}{x \cdot x}\right)}}{t}}\\ \mathbf{elif}\;t \leq 10^{-198}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-69}:\\ \;\;\;\;\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 5: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{\ell}\\ t_2 := t_1 \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ t_3 := \sqrt{\frac{x + 1}{-1 + x}}\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{t_3}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.28 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{t_1}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot t_3}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) l))
        (t_2 (* t_1 (* t (pow (/ 2.0 x) -0.5))))
        (t_3 (sqrt (/ (+ x 1.0) (+ -1.0 x)))))
   (if (<= t -4.4e-72)
     (/ -1.0 t_3)
     (if (<= t -1.6e-109)
       t_2
       (if (<= t -1.28e-185)
         -1.0
         (if (<= t 3.4e-198)
           t_2
           (if (<= t 1.02e-122)
             1.0
             (if (<= t 2.7e-70)
               (* t (/ t_1 (sqrt (/ 2.0 x))))
               (* t (/ (sqrt 2.0) (* (* t (sqrt 2.0)) t_3)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) / l;
	double t_2 = t_1 * (t * pow((2.0 / x), -0.5));
	double t_3 = sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -4.4e-72) {
		tmp = -1.0 / t_3;
	} else if (t <= -1.6e-109) {
		tmp = t_2;
	} else if (t <= -1.28e-185) {
		tmp = -1.0;
	} else if (t <= 3.4e-198) {
		tmp = t_2;
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 2.7e-70) {
		tmp = t * (t_1 / sqrt((2.0 / x)));
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * t_3));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(2.0d0) / l
    t_2 = t_1 * (t * ((2.0d0 / x) ** (-0.5d0)))
    t_3 = sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    if (t <= (-4.4d-72)) then
        tmp = (-1.0d0) / t_3
    else if (t <= (-1.6d-109)) then
        tmp = t_2
    else if (t <= (-1.28d-185)) then
        tmp = -1.0d0
    else if (t <= 3.4d-198) then
        tmp = t_2
    else if (t <= 1.02d-122) then
        tmp = 1.0d0
    else if (t <= 2.7d-70) then
        tmp = t * (t_1 / sqrt((2.0d0 / x)))
    else
        tmp = t * (sqrt(2.0d0) / ((t * sqrt(2.0d0)) * t_3))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(2.0) / l;
	double t_2 = t_1 * (t * Math.pow((2.0 / x), -0.5));
	double t_3 = Math.sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -4.4e-72) {
		tmp = -1.0 / t_3;
	} else if (t <= -1.6e-109) {
		tmp = t_2;
	} else if (t <= -1.28e-185) {
		tmp = -1.0;
	} else if (t <= 3.4e-198) {
		tmp = t_2;
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 2.7e-70) {
		tmp = t * (t_1 / Math.sqrt((2.0 / x)));
	} else {
		tmp = t * (Math.sqrt(2.0) / ((t * Math.sqrt(2.0)) * t_3));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(2.0) / l
	t_2 = t_1 * (t * math.pow((2.0 / x), -0.5))
	t_3 = math.sqrt(((x + 1.0) / (-1.0 + x)))
	tmp = 0
	if t <= -4.4e-72:
		tmp = -1.0 / t_3
	elif t <= -1.6e-109:
		tmp = t_2
	elif t <= -1.28e-185:
		tmp = -1.0
	elif t <= 3.4e-198:
		tmp = t_2
	elif t <= 1.02e-122:
		tmp = 1.0
	elif t <= 2.7e-70:
		tmp = t * (t_1 / math.sqrt((2.0 / x)))
	else:
		tmp = t * (math.sqrt(2.0) / ((t * math.sqrt(2.0)) * t_3))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) / l)
	t_2 = Float64(t_1 * Float64(t * (Float64(2.0 / x) ^ -0.5)))
	t_3 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	tmp = 0.0
	if (t <= -4.4e-72)
		tmp = Float64(-1.0 / t_3);
	elseif (t <= -1.6e-109)
		tmp = t_2;
	elseif (t <= -1.28e-185)
		tmp = -1.0;
	elseif (t <= 3.4e-198)
		tmp = t_2;
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 2.7e-70)
		tmp = Float64(t * Float64(t_1 / sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * t_3)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) / l;
	t_2 = t_1 * (t * ((2.0 / x) ^ -0.5));
	t_3 = sqrt(((x + 1.0) / (-1.0 + x)));
	tmp = 0.0;
	if (t <= -4.4e-72)
		tmp = -1.0 / t_3;
	elseif (t <= -1.6e-109)
		tmp = t_2;
	elseif (t <= -1.28e-185)
		tmp = -1.0;
	elseif (t <= 3.4e-198)
		tmp = t_2;
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 2.7e-70)
		tmp = t * (t_1 / sqrt((2.0 / x)));
	else
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * t_3));
	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[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.4e-72], N[(-1.0 / t$95$3), $MachinePrecision], If[LessEqual[t, -1.6e-109], t$95$2, If[LessEqual[t, -1.28e-185], -1.0, If[LessEqual[t, 3.4e-198], t$95$2, If[LessEqual[t, 1.02e-122], 1.0, If[LessEqual[t, 2.7e-70], N[(t * N[(t$95$1 / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.40000000000000005e-72

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. +-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. metadata-eval 84.4%

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

      6. +-commutative 84.4%

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

   6. Simplified 84.4%

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

      1. frac-2neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. times-frac 84.4%

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

      6. neg-mul-1 84.4%

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

      7. *-commutative 84.4%

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

      8. distribute-rgt-neg-in 84.4%

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

      9. metadata-eval 84.4%

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

      10. +-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

   10. Simplified 84.4%

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

   if -4.40000000000000005e-72 < t < -1.6000000000000001e-109 or -1.28e-185 < t < 3.3999999999999998e-198

   1. Initial program 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in l around inf 0.9%

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

   5. Taylor expanded in x around inf 41.9%

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

      1. associate-/r* 41.9%

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

      2. div-inv 41.9%

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

      3. associate-/r/ 42.3%

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

      4. pow1/2 42.3%

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

      5. pow-flip 42.3%

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

      6. metadata-eval 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. associate-*l* 48.2%

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

   9. Simplified 48.2%

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

   if -1.6000000000000001e-109 < t < -1.28e-185

   1. Initial program 31.6%

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

   3. Simplified 31.5%

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

   4. Taylor expanded in t around -inf 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. +-commutative 93.6%

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

      4. sub-neg 93.6%

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

      5. metadata-eval 93.6%

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

      6. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in x around inf 93.6%

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

   if 3.3999999999999998e-198 < t < 1.02000000000000002e-122

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

      1. rewrite-binary64/binary32 48.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 48.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 86.7%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 86.7%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 86.7%

         \[\leadsto 1 \]

   10. Applied egg-rr 86.7%

       \[\leadsto 1 \]

   if 1.02000000000000002e-122 < t < 2.7000000000000001e-70

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in l around inf 1.7%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*r/ 38.6%

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

      3. *-commutative 38.6%

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

      4. sqrt-div 38.6%

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

      5. associate-*r/ 38.5%

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

      6. un-div-inv 38.5%

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

      7. metadata-eval 38.5%

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

      8. sqrt-div 38.5%

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

      9. *-commutative 38.5%

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

      10. *-commutative 38.5%

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

      11. associate-/r/ 45.8%

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

   7. Applied egg-rr 45.8%

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

   if 2.7000000000000001e-70 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in t around inf 88.2%

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

      1. +-commutative 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -1.28 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-198}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 6: 77.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ t_2 := \sqrt{\frac{x + 1}{-1 + x}}\\ \mathbf{if}\;t \leq -4.45 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{t_2}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot t_2}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ (sqrt 2.0) l) (* t (pow (/ 2.0 x) -0.5))))
        (t_2 (sqrt (/ (+ x 1.0) (+ -1.0 x)))))
   (if (<= t -4.45e-72)
     (/ -1.0 t_2)
     (if (<= t -5.5e-113)
       t_1
       (if (<= t -4.4e-185)
         -1.0
         (if (<= t 6e-197)
           t_1
           (if (<= t 1.02e-122)
             1.0
             (if (<= t 2.35e-70)
               (*
                (/ (pow 2.0 0.25) (/ 1.0 t))
                (/ (pow 2.0 0.25) (* l (sqrt (/ 2.0 x)))))
               (* t (/ (sqrt 2.0) (* (* t (sqrt 2.0)) t_2)))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (sqrt(2.0) / l) * (t * pow((2.0 / x), -0.5));
	double t_2 = sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -4.45e-72) {
		tmp = -1.0 / t_2;
	} else if (t <= -5.5e-113) {
		tmp = t_1;
	} else if (t <= -4.4e-185) {
		tmp = -1.0;
	} else if (t <= 6e-197) {
		tmp = t_1;
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 2.35e-70) {
		tmp = (pow(2.0, 0.25) / (1.0 / t)) * (pow(2.0, 0.25) / (l * sqrt((2.0 / x))));
	} else {
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * t_2));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (sqrt(2.0d0) / l) * (t * ((2.0d0 / x) ** (-0.5d0)))
    t_2 = sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    if (t <= (-4.45d-72)) then
        tmp = (-1.0d0) / t_2
    else if (t <= (-5.5d-113)) then
        tmp = t_1
    else if (t <= (-4.4d-185)) then
        tmp = -1.0d0
    else if (t <= 6d-197) then
        tmp = t_1
    else if (t <= 1.02d-122) then
        tmp = 1.0d0
    else if (t <= 2.35d-70) then
        tmp = ((2.0d0 ** 0.25d0) / (1.0d0 / t)) * ((2.0d0 ** 0.25d0) / (l * sqrt((2.0d0 / x))))
    else
        tmp = t * (sqrt(2.0d0) / ((t * sqrt(2.0d0)) * t_2))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = (Math.sqrt(2.0) / l) * (t * Math.pow((2.0 / x), -0.5));
	double t_2 = Math.sqrt(((x + 1.0) / (-1.0 + x)));
	double tmp;
	if (t <= -4.45e-72) {
		tmp = -1.0 / t_2;
	} else if (t <= -5.5e-113) {
		tmp = t_1;
	} else if (t <= -4.4e-185) {
		tmp = -1.0;
	} else if (t <= 6e-197) {
		tmp = t_1;
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 2.35e-70) {
		tmp = (Math.pow(2.0, 0.25) / (1.0 / t)) * (Math.pow(2.0, 0.25) / (l * Math.sqrt((2.0 / x))));
	} else {
		tmp = t * (Math.sqrt(2.0) / ((t * Math.sqrt(2.0)) * t_2));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (math.sqrt(2.0) / l) * (t * math.pow((2.0 / x), -0.5))
	t_2 = math.sqrt(((x + 1.0) / (-1.0 + x)))
	tmp = 0
	if t <= -4.45e-72:
		tmp = -1.0 / t_2
	elif t <= -5.5e-113:
		tmp = t_1
	elif t <= -4.4e-185:
		tmp = -1.0
	elif t <= 6e-197:
		tmp = t_1
	elif t <= 1.02e-122:
		tmp = 1.0
	elif t <= 2.35e-70:
		tmp = (math.pow(2.0, 0.25) / (1.0 / t)) * (math.pow(2.0, 0.25) / (l * math.sqrt((2.0 / x))))
	else:
		tmp = t * (math.sqrt(2.0) / ((t * math.sqrt(2.0)) * t_2))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(sqrt(2.0) / l) * Float64(t * (Float64(2.0 / x) ^ -0.5)))
	t_2 = sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x)))
	tmp = 0.0
	if (t <= -4.45e-72)
		tmp = Float64(-1.0 / t_2);
	elseif (t <= -5.5e-113)
		tmp = t_1;
	elseif (t <= -4.4e-185)
		tmp = -1.0;
	elseif (t <= 6e-197)
		tmp = t_1;
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 2.35e-70)
		tmp = Float64(Float64((2.0 ^ 0.25) / Float64(1.0 / t)) * Float64((2.0 ^ 0.25) / Float64(l * sqrt(Float64(2.0 / x)))));
	else
		tmp = Float64(t * Float64(sqrt(2.0) / Float64(Float64(t * sqrt(2.0)) * t_2)));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (sqrt(2.0) / l) * (t * ((2.0 / x) ^ -0.5));
	t_2 = sqrt(((x + 1.0) / (-1.0 + x)));
	tmp = 0.0;
	if (t <= -4.45e-72)
		tmp = -1.0 / t_2;
	elseif (t <= -5.5e-113)
		tmp = t_1;
	elseif (t <= -4.4e-185)
		tmp = -1.0;
	elseif (t <= 6e-197)
		tmp = t_1;
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 2.35e-70)
		tmp = ((2.0 ^ 0.25) / (1.0 / t)) * ((2.0 ^ 0.25) / (l * sqrt((2.0 / x))));
	else
		tmp = t * (sqrt(2.0) / ((t * sqrt(2.0)) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.45e-72], N[(-1.0 / t$95$2), $MachinePrecision], If[LessEqual[t, -5.5e-113], t$95$1, If[LessEqual[t, -4.4e-185], -1.0, If[LessEqual[t, 6e-197], t$95$1, If[LessEqual[t, 1.02e-122], 1.0, If[LessEqual[t, 2.35e-70], N[(N[(N[Power[2.0, 0.25], $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] / N[(l * N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.4499999999999999e-72

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. +-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. metadata-eval 84.4%

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

      6. +-commutative 84.4%

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

   6. Simplified 84.4%

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

      1. frac-2neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. times-frac 84.4%

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

      6. neg-mul-1 84.4%

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

      7. *-commutative 84.4%

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

      8. distribute-rgt-neg-in 84.4%

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

      9. metadata-eval 84.4%

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

      10. +-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

   10. Simplified 84.4%

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

   if -4.4499999999999999e-72 < t < -5.50000000000000053e-113 or -4.4000000000000001e-185 < t < 6.00000000000000051e-197

   1. Initial program 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in l around inf 0.9%

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

   5. Taylor expanded in x around inf 41.9%

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

      1. associate-/r* 41.9%

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

      2. div-inv 41.9%

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

      3. associate-/r/ 42.3%

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

      4. pow1/2 42.3%

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

      5. pow-flip 42.3%

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

      6. metadata-eval 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. associate-*l* 48.2%

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

   9. Simplified 48.2%

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

   if -5.50000000000000053e-113 < t < -4.4000000000000001e-185

   1. Initial program 31.6%

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

   3. Simplified 31.5%

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

   4. Taylor expanded in t around -inf 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. +-commutative 93.6%

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

      4. sub-neg 93.6%

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

      5. metadata-eval 93.6%

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

      6. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in x around inf 93.6%

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

   if 6.00000000000000051e-197 < t < 1.02000000000000002e-122

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

      1. rewrite-binary64/binary32 48.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 48.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 86.7%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 86.7%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 86.7%

         \[\leadsto 1 \]

   10. Applied egg-rr 86.7%

       \[\leadsto 1 \]

   if 1.02000000000000002e-122 < t < 2.3499999999999999e-70

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in l around inf 1.7%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. pow1/2 38.8%

         \[\leadsto \frac{\color{blue}{{2}^{0.5}}}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}} \]

      2. sqr-pow 38.6%

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

      3. *-commutative 38.6%

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

      4. div-inv 38.6%

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

      5. associate-*r* 38.6%

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

      6. *-commutative 38.6%

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

      7. sqrt-div 38.4%

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

      8. associate-*r/ 38.2%

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

      9. un-div-inv 38.2%

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

      10. metadata-eval 38.2%

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

      11. sqrt-div 38.5%

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

      12. *-commutative 38.5%

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

      13. times-frac 45.6%

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

      14. metadata-eval 45.6%

          \[\leadsto \frac{{2}^{\color{blue}{0.25}}}{\frac{1}{t}} \cdot \frac{{2}^{\left(\frac{0.5}{2}\right)}}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}} \]

      15. metadata-eval 45.6%

          \[\leadsto \frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{\color{blue}{0.25}}}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}} \]

      16. sqrt-div 45.5%

          \[\leadsto \frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}} \]

   7. Applied egg-rr 45.8%

      \[\leadsto \color{blue}{\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}} \]

   if 2.3499999999999999e-70 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in t around inf 88.2%

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

      1. +-commutative 88.2%

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

      2. sub-neg 88.2%

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

      3. metadata-eval 88.2%

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

      4. +-commutative 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.45 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-70}:\\ \;\;\;\;\frac{{2}^{0.25}}{\frac{1}{t}} \cdot \frac{{2}^{0.25}}{\ell \cdot \sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{\sqrt{2}}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}\\ \end{array} \]

Alternative 7: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{if}\;t \leq -1.16 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ (sqrt 2.0) l) (/ t (sqrt (/ 2.0 x))))))
   (if (<= t -1.16e-69)
     (/ -1.0 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
     (if (<= t -2.3e-114)
       t_1
       (if (<= t -6.2e-184)
         -1.0
         (if (<= t 2.2e-197)
           t_1
           (if (<= t 1e-122)
             1.0
             (if (<= t 1.9e-70)
               t_1
               (/ (sqrt 2.0) (sqrt (+ 2.0 (/ 4.0 x))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (sqrt(2.0) / l) * (t / sqrt((2.0 / x)));
	double tmp;
	if (t <= -1.16e-69) {
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -2.3e-114) {
		tmp = t_1;
	} else if (t <= -6.2e-184) {
		tmp = -1.0;
	} else if (t <= 2.2e-197) {
		tmp = t_1;
	} else if (t <= 1e-122) {
		tmp = 1.0;
	} else if (t <= 1.9e-70) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (sqrt(2.0d0) / l) * (t / sqrt((2.0d0 / x)))
    if (t <= (-1.16d-69)) then
        tmp = (-1.0d0) / sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    else if (t <= (-2.3d-114)) then
        tmp = t_1
    else if (t <= (-6.2d-184)) then
        tmp = -1.0d0
    else if (t <= 2.2d-197) then
        tmp = t_1
    else if (t <= 1d-122) then
        tmp = 1.0d0
    else if (t <= 1.9d-70) then
        tmp = t_1
    else
        tmp = sqrt(2.0d0) / sqrt((2.0d0 + (4.0d0 / x)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = (Math.sqrt(2.0) / l) * (t / Math.sqrt((2.0 / x)));
	double tmp;
	if (t <= -1.16e-69) {
		tmp = -1.0 / Math.sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -2.3e-114) {
		tmp = t_1;
	} else if (t <= -6.2e-184) {
		tmp = -1.0;
	} else if (t <= 2.2e-197) {
		tmp = t_1;
	} else if (t <= 1e-122) {
		tmp = 1.0;
	} else if (t <= 1.9e-70) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (math.sqrt(2.0) / l) * (t / math.sqrt((2.0 / x)))
	tmp = 0
	if t <= -1.16e-69:
		tmp = -1.0 / math.sqrt(((x + 1.0) / (-1.0 + x)))
	elif t <= -2.3e-114:
		tmp = t_1
	elif t <= -6.2e-184:
		tmp = -1.0
	elif t <= 2.2e-197:
		tmp = t_1
	elif t <= 1e-122:
		tmp = 1.0
	elif t <= 1.9e-70:
		tmp = t_1
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 + (4.0 / x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(sqrt(2.0) / l) * Float64(t / sqrt(Float64(2.0 / x))))
	tmp = 0.0
	if (t <= -1.16e-69)
		tmp = Float64(-1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))));
	elseif (t <= -2.3e-114)
		tmp = t_1;
	elseif (t <= -6.2e-184)
		tmp = -1.0;
	elseif (t <= 2.2e-197)
		tmp = t_1;
	elseif (t <= 1e-122)
		tmp = 1.0;
	elseif (t <= 1.9e-70)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 + Float64(4.0 / x))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (sqrt(2.0) / l) * (t / sqrt((2.0 / x)));
	tmp = 0.0;
	if (t <= -1.16e-69)
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	elseif (t <= -2.3e-114)
		tmp = t_1;
	elseif (t <= -6.2e-184)
		tmp = -1.0;
	elseif (t <= 2.2e-197)
		tmp = t_1;
	elseif (t <= 1e-122)
		tmp = 1.0;
	elseif (t <= 1.9e-70)
		tmp = t_1;
	else
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.16e-69], N[(-1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.3e-114], t$95$1, If[LessEqual[t, -6.2e-184], -1.0, If[LessEqual[t, 2.2e-197], t$95$1, If[LessEqual[t, 1e-122], 1.0, If[LessEqual[t, 1.9e-70], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.15999999999999989e-69

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. +-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. metadata-eval 84.4%

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

      6. +-commutative 84.4%

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

   6. Simplified 84.4%

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

      1. frac-2neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. times-frac 84.4%

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

      6. neg-mul-1 84.4%

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

      7. *-commutative 84.4%

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

      8. distribute-rgt-neg-in 84.4%

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

      9. metadata-eval 84.4%

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

      10. +-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

   10. Simplified 84.4%

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

   if -1.15999999999999989e-69 < t < -2.2999999999999999e-114 or -6.2000000000000004e-184 < t < 2.2e-197 or 1.00000000000000006e-122 < t < 1.8999999999999999e-70

   1. Initial program 9.8%

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

   3. Simplified 9.8%

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

   4. Taylor expanded in l around inf 1.0%

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

   5. Taylor expanded in x around inf 41.4%

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

      1. associate-/r* 41.3%

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

      2. div-inv 41.4%

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

      3. times-frac 41.4%

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

      4. *-commutative 41.4%

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

      5. times-frac 41.4%

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

      6. clear-num 41.9%

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

      7. times-frac 47.7%

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

      8. *-commutative 47.7%

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

      9. times-frac 47.8%

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

   7. Applied egg-rr 47.8%

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

   if -2.2999999999999999e-114 < t < -6.2000000000000004e-184

   1. Initial program 31.6%

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

   3. Simplified 31.5%

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

   4. Taylor expanded in t around -inf 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. +-commutative 93.6%

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

      4. sub-neg 93.6%

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

      5. metadata-eval 93.6%

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

      6. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in x around inf 93.6%

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

   if 2.2e-197 < t < 1.00000000000000006e-122

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

      1. rewrite-binary64/binary32 48.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 48.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 86.7%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 86.7%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 86.7%

         \[\leadsto 1 \]

   10. Applied egg-rr 86.7%

       \[\leadsto 1 \]

   if 1.8999999999999999e-70 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in x around inf 48.0%

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

      1. fma-def 48.0%

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

      2. unpow2 48.0%

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

      3. fma-def 48.0%

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

      4. unpow2 48.0%

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

      5. unpow2 48.0%

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

      6. associate-*r/ 48.0%

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

      7. mul-1-neg 48.0%

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

      8. unpow2 48.0%

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

      9. fma-def 48.0%

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

      10. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in t around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. metadata-eval 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in t around 0 87.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + 4 \cdot \frac{1}{x}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 87.8%

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

       2. metadata-eval 87.8%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2 + \frac{\color{blue}{4}}{x}}} \]

   12. Simplified 87.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + \frac{4}{x}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 8: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{\ell}\\ t_2 := \sqrt{\frac{2}{x}}\\ t_3 := t_1 \cdot \frac{t}{t_2}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-181}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-197}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{t_1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) l)) (t_2 (sqrt (/ 2.0 x))) (t_3 (* t_1 (/ t t_2))))
   (if (<= t -3e-67)
     (/ -1.0 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
     (if (<= t -7.5e-110)
       t_3
       (if (<= t -7.2e-181)
         -1.0
         (if (<= t 2.9e-197)
           t_3
           (if (<= t 1.02e-122)
             1.0
             (if (<= t 1.75e-70)
               (* t (/ t_1 t_2))
               (/ (sqrt 2.0) (sqrt (+ 2.0 (/ 4.0 x))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) / l;
	double t_2 = sqrt((2.0 / x));
	double t_3 = t_1 * (t / t_2);
	double tmp;
	if (t <= -3e-67) {
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -7.5e-110) {
		tmp = t_3;
	} else if (t <= -7.2e-181) {
		tmp = -1.0;
	} else if (t <= 2.9e-197) {
		tmp = t_3;
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 1.75e-70) {
		tmp = t * (t_1 / t_2);
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = sqrt(2.0d0) / l
    t_2 = sqrt((2.0d0 / x))
    t_3 = t_1 * (t / t_2)
    if (t <= (-3d-67)) then
        tmp = (-1.0d0) / sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    else if (t <= (-7.5d-110)) then
        tmp = t_3
    else if (t <= (-7.2d-181)) then
        tmp = -1.0d0
    else if (t <= 2.9d-197) then
        tmp = t_3
    else if (t <= 1.02d-122) then
        tmp = 1.0d0
    else if (t <= 1.75d-70) then
        tmp = t * (t_1 / t_2)
    else
        tmp = sqrt(2.0d0) / sqrt((2.0d0 + (4.0d0 / x)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(2.0) / l;
	double t_2 = Math.sqrt((2.0 / x));
	double t_3 = t_1 * (t / t_2);
	double tmp;
	if (t <= -3e-67) {
		tmp = -1.0 / Math.sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -7.5e-110) {
		tmp = t_3;
	} else if (t <= -7.2e-181) {
		tmp = -1.0;
	} else if (t <= 2.9e-197) {
		tmp = t_3;
	} else if (t <= 1.02e-122) {
		tmp = 1.0;
	} else if (t <= 1.75e-70) {
		tmp = t * (t_1 / t_2);
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(2.0) / l
	t_2 = math.sqrt((2.0 / x))
	t_3 = t_1 * (t / t_2)
	tmp = 0
	if t <= -3e-67:
		tmp = -1.0 / math.sqrt(((x + 1.0) / (-1.0 + x)))
	elif t <= -7.5e-110:
		tmp = t_3
	elif t <= -7.2e-181:
		tmp = -1.0
	elif t <= 2.9e-197:
		tmp = t_3
	elif t <= 1.02e-122:
		tmp = 1.0
	elif t <= 1.75e-70:
		tmp = t * (t_1 / t_2)
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 + (4.0 / x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) / l)
	t_2 = sqrt(Float64(2.0 / x))
	t_3 = Float64(t_1 * Float64(t / t_2))
	tmp = 0.0
	if (t <= -3e-67)
		tmp = Float64(-1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))));
	elseif (t <= -7.5e-110)
		tmp = t_3;
	elseif (t <= -7.2e-181)
		tmp = -1.0;
	elseif (t <= 2.9e-197)
		tmp = t_3;
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 1.75e-70)
		tmp = Float64(t * Float64(t_1 / t_2));
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 + Float64(4.0 / x))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) / l;
	t_2 = sqrt((2.0 / x));
	t_3 = t_1 * (t / t_2);
	tmp = 0.0;
	if (t <= -3e-67)
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	elseif (t <= -7.5e-110)
		tmp = t_3;
	elseif (t <= -7.2e-181)
		tmp = -1.0;
	elseif (t <= 2.9e-197)
		tmp = t_3;
	elseif (t <= 1.02e-122)
		tmp = 1.0;
	elseif (t <= 1.75e-70)
		tmp = t * (t_1 / t_2);
	else
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[(t / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e-67], N[(-1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.5e-110], t$95$3, If[LessEqual[t, -7.2e-181], -1.0, If[LessEqual[t, 2.9e-197], t$95$3, If[LessEqual[t, 1.02e-122], 1.0, If[LessEqual[t, 1.75e-70], N[(t * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -3.00000000000000032e-67

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. +-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. metadata-eval 84.4%

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

      6. +-commutative 84.4%

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

   6. Simplified 84.4%

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

      1. frac-2neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. times-frac 84.4%

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

      6. neg-mul-1 84.4%

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

      7. *-commutative 84.4%

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

      8. distribute-rgt-neg-in 84.4%

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

      9. metadata-eval 84.4%

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

      10. +-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

   10. Simplified 84.4%

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

   if -3.00000000000000032e-67 < t < -7.50000000000000053e-110 or -7.1999999999999998e-181 < t < 2.90000000000000023e-197

   1. Initial program 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in l around inf 0.9%

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

   5. Taylor expanded in x around inf 41.9%

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

      1. associate-/r* 41.9%

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

      2. div-inv 41.9%

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

      3. times-frac 41.9%

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

      4. *-commutative 41.9%

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

      5. times-frac 41.9%

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

      6. clear-num 42.5%

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

      7. times-frac 48.1%

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

      8. *-commutative 48.1%

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

      9. times-frac 48.3%

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

   7. Applied egg-rr 48.3%

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

   if -7.50000000000000053e-110 < t < -7.1999999999999998e-181

   1. Initial program 31.6%

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

   3. Simplified 31.5%

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

   4. Taylor expanded in t around -inf 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. +-commutative 93.6%

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

      4. sub-neg 93.6%

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

      5. metadata-eval 93.6%

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

      6. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in x around inf 93.6%

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

   if 2.90000000000000023e-197 < t < 1.02000000000000002e-122

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

      1. rewrite-binary64/binary32 48.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 48.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 86.7%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 86.7%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 86.7%

         \[\leadsto 1 \]

   10. Applied egg-rr 86.7%

       \[\leadsto 1 \]

   if 1.02000000000000002e-122 < t < 1.74999999999999987e-70

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in l around inf 1.7%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*r/ 38.6%

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

      3. *-commutative 38.6%

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

      4. sqrt-div 38.6%

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

      5. associate-*r/ 38.5%

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

      6. un-div-inv 38.5%

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

      7. metadata-eval 38.5%

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

      8. sqrt-div 38.5%

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

      9. *-commutative 38.5%

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

      10. *-commutative 38.5%

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

      11. associate-/r/ 45.8%

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

   7. Applied egg-rr 45.8%

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

   if 1.74999999999999987e-70 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in x around inf 48.0%

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

      1. fma-def 48.0%

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

      2. unpow2 48.0%

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

      3. fma-def 48.0%

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

      4. unpow2 48.0%

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

      5. unpow2 48.0%

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

      6. associate-*r/ 48.0%

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

      7. mul-1-neg 48.0%

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

      8. unpow2 48.0%

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

      9. fma-def 48.0%

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

      10. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in t around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. metadata-eval 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in t around 0 87.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + 4 \cdot \frac{1}{x}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 87.8%

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

       2. metadata-eval 87.8%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2 + \frac{\color{blue}{4}}{x}}} \]

   12. Simplified 87.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + \frac{4}{x}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-110}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-181}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \frac{t}{\sqrt{\frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-122}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 9: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{2}}{\ell}\\ t_2 := t_1 \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.7 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{-182}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-123}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{t_1}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (/ (sqrt 2.0) l)) (t_2 (* t_1 (* t (pow (/ 2.0 x) -0.5)))))
   (if (<= t -4.4e-72)
     (/ -1.0 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
     (if (<= t -7.7e-109)
       t_2
       (if (<= t -6.8e-182)
         -1.0
         (if (<= t 7.5e-197)
           t_2
           (if (<= t 9.5e-123)
             1.0
             (if (<= t 1.9e-70)
               (* t (/ t_1 (sqrt (/ 2.0 x))))
               (/ (sqrt 2.0) (sqrt (+ 2.0 (/ 4.0 x))))))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = sqrt(2.0) / l;
	double t_2 = t_1 * (t * pow((2.0 / x), -0.5));
	double tmp;
	if (t <= -4.4e-72) {
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -7.7e-109) {
		tmp = t_2;
	} else if (t <= -6.8e-182) {
		tmp = -1.0;
	} else if (t <= 7.5e-197) {
		tmp = t_2;
	} else if (t <= 9.5e-123) {
		tmp = 1.0;
	} else if (t <= 1.9e-70) {
		tmp = t * (t_1 / sqrt((2.0 / x)));
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(2.0d0) / l
    t_2 = t_1 * (t * ((2.0d0 / x) ** (-0.5d0)))
    if (t <= (-4.4d-72)) then
        tmp = (-1.0d0) / sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    else if (t <= (-7.7d-109)) then
        tmp = t_2
    else if (t <= (-6.8d-182)) then
        tmp = -1.0d0
    else if (t <= 7.5d-197) then
        tmp = t_2
    else if (t <= 9.5d-123) then
        tmp = 1.0d0
    else if (t <= 1.9d-70) then
        tmp = t * (t_1 / sqrt((2.0d0 / x)))
    else
        tmp = sqrt(2.0d0) / sqrt((2.0d0 + (4.0d0 / x)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = Math.sqrt(2.0) / l;
	double t_2 = t_1 * (t * Math.pow((2.0 / x), -0.5));
	double tmp;
	if (t <= -4.4e-72) {
		tmp = -1.0 / Math.sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -7.7e-109) {
		tmp = t_2;
	} else if (t <= -6.8e-182) {
		tmp = -1.0;
	} else if (t <= 7.5e-197) {
		tmp = t_2;
	} else if (t <= 9.5e-123) {
		tmp = 1.0;
	} else if (t <= 1.9e-70) {
		tmp = t * (t_1 / Math.sqrt((2.0 / x)));
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = math.sqrt(2.0) / l
	t_2 = t_1 * (t * math.pow((2.0 / x), -0.5))
	tmp = 0
	if t <= -4.4e-72:
		tmp = -1.0 / math.sqrt(((x + 1.0) / (-1.0 + x)))
	elif t <= -7.7e-109:
		tmp = t_2
	elif t <= -6.8e-182:
		tmp = -1.0
	elif t <= 7.5e-197:
		tmp = t_2
	elif t <= 9.5e-123:
		tmp = 1.0
	elif t <= 1.9e-70:
		tmp = t * (t_1 / math.sqrt((2.0 / x)))
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 + (4.0 / x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(sqrt(2.0) / l)
	t_2 = Float64(t_1 * Float64(t * (Float64(2.0 / x) ^ -0.5)))
	tmp = 0.0
	if (t <= -4.4e-72)
		tmp = Float64(-1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))));
	elseif (t <= -7.7e-109)
		tmp = t_2;
	elseif (t <= -6.8e-182)
		tmp = -1.0;
	elseif (t <= 7.5e-197)
		tmp = t_2;
	elseif (t <= 9.5e-123)
		tmp = 1.0;
	elseif (t <= 1.9e-70)
		tmp = Float64(t * Float64(t_1 / sqrt(Float64(2.0 / x))));
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 + Float64(4.0 / x))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = sqrt(2.0) / l;
	t_2 = t_1 * (t * ((2.0 / x) ^ -0.5));
	tmp = 0.0;
	if (t <= -4.4e-72)
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	elseif (t <= -7.7e-109)
		tmp = t_2;
	elseif (t <= -6.8e-182)
		tmp = -1.0;
	elseif (t <= 7.5e-197)
		tmp = t_2;
	elseif (t <= 9.5e-123)
		tmp = 1.0;
	elseif (t <= 1.9e-70)
		tmp = t * (t_1 / sqrt((2.0 / x)));
	else
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := Block[{t$95$1 = N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t * N[Power[N[(2.0 / x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.4e-72], N[(-1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.7e-109], t$95$2, If[LessEqual[t, -6.8e-182], -1.0, If[LessEqual[t, 7.5e-197], t$95$2, If[LessEqual[t, 9.5e-123], 1.0, If[LessEqual[t, 1.9e-70], N[(t * N[(t$95$1 / N[Sqrt[N[(2.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -4.40000000000000005e-72

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. +-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. metadata-eval 84.4%

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

      6. +-commutative 84.4%

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

   6. Simplified 84.4%

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

      1. frac-2neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. times-frac 84.4%

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

      6. neg-mul-1 84.4%

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

      7. *-commutative 84.4%

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

      8. distribute-rgt-neg-in 84.4%

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

      9. metadata-eval 84.4%

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

      10. +-commutative 84.4%

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

   8. Applied egg-rr 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-rgt-identity 84.4%

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

      3. *-inverses 84.4%

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

      4. metadata-eval 84.4%

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

   10. Simplified 84.4%

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

   if -4.40000000000000005e-72 < t < -7.70000000000000025e-109 or -6.79999999999999979e-182 < t < 7.5e-197

   1. Initial program 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in l around inf 0.9%

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

   5. Taylor expanded in x around inf 41.9%

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

      1. associate-/r* 41.9%

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

      2. div-inv 41.9%

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

      3. associate-/r/ 42.3%

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

      4. pow1/2 42.3%

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

      5. pow-flip 42.3%

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

      6. metadata-eval 42.3%

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

   7. Applied egg-rr 42.3%

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

      1. associate-*l* 48.2%

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

   9. Simplified 48.2%

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

   if -7.70000000000000025e-109 < t < -6.79999999999999979e-182

   1. Initial program 31.6%

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

   3. Simplified 31.5%

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

   4. Taylor expanded in t around -inf 93.6%

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

      1. associate-*r* 93.6%

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

      2. neg-mul-1 93.6%

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

      3. +-commutative 93.6%

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

      4. sub-neg 93.6%

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

      5. metadata-eval 93.6%

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

      6. +-commutative 93.6%

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

   6. Simplified 93.6%

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

   7. Taylor expanded in x around inf 93.6%

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

   if 7.5e-197 < t < 9.5000000000000002e-123

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in x around inf 85.4%

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

      1. *-commutative 85.4%

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

   6. Simplified 85.4%

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

      1. rewrite-binary64/binary32 48.0%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 48.0%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 86.7%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 86.7%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 86.7%

         \[\leadsto 1 \]

   10. Applied egg-rr 86.7%

       \[\leadsto 1 \]

   if 9.5000000000000002e-123 < t < 1.8999999999999999e-70

   1. Initial program 23.0%

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

   3. Simplified 23.0%

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

   4. Taylor expanded in l around inf 1.7%

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

   5. Taylor expanded in x around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. associate-*r/ 38.6%

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

      3. *-commutative 38.6%

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

      4. sqrt-div 38.6%

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

      5. associate-*r/ 38.5%

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

      6. un-div-inv 38.5%

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

      7. metadata-eval 38.5%

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

      8. sqrt-div 38.5%

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

      9. *-commutative 38.5%

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

      10. *-commutative 38.5%

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

      11. associate-/r/ 45.8%

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

   7. Applied egg-rr 45.8%

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

   if 1.8999999999999999e-70 < t

   1. Initial program 42.1%

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

   3. Simplified 42.1%

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

   4. Taylor expanded in x around inf 48.0%

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

      1. fma-def 48.0%

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

      2. unpow2 48.0%

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

      3. fma-def 48.0%

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

      4. unpow2 48.0%

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

      5. unpow2 48.0%

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

      6. associate-*r/ 48.0%

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

      7. mul-1-neg 48.0%

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

      8. unpow2 48.0%

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

      9. fma-def 48.0%

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

      10. unpow2 48.0%

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

   6. Simplified 48.0%

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

   7. Taylor expanded in t around inf 87.7%

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

      1. associate-*r/ 87.7%

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

      2. metadata-eval 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in t around 0 87.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + 4 \cdot \frac{1}{x}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 87.8%

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

       2. metadata-eval 87.8%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2 + \frac{\color{blue}{4}}{x}}} \]

   12. Simplified 87.8%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + \frac{4}{x}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.7 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{-182}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{\sqrt{2}}{\ell} \cdot \left(t \cdot {\left(\frac{2}{x}\right)}^{-0.5}\right)\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-123}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{\frac{\sqrt{2}}{\ell}}{\sqrt{\frac{2}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 10: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -4.4e-72)
   (/ -1.0 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
   (if (<= t -1.65e-109)
     (* (/ t l) (sqrt x))
     (if (<= t -6.5e-184)
       -1.0
       (if (<= t 1.06e-197)
         (/ (/ t l) (pow x -0.5))
         (/ (sqrt 2.0) (sqrt (+ 2.0 (/ 4.0 x)))))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.4e-72) {
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -1.65e-109) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= -6.5e-184) {
		tmp = -1.0;
	} else if (t <= 1.06e-197) {
		tmp = (t / l) / pow(x, -0.5);
	} else {
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.4d-72)) then
        tmp = (-1.0d0) / sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    else if (t <= (-1.65d-109)) then
        tmp = (t / l) * sqrt(x)
    else if (t <= (-6.5d-184)) then
        tmp = -1.0d0
    else if (t <= 1.06d-197) then
        tmp = (t / l) / (x ** (-0.5d0))
    else
        tmp = sqrt(2.0d0) / sqrt((2.0d0 + (4.0d0 / x)))
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.4e-72) {
		tmp = -1.0 / Math.sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -1.65e-109) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= -6.5e-184) {
		tmp = -1.0;
	} else if (t <= 1.06e-197) {
		tmp = (t / l) / Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt(2.0) / Math.sqrt((2.0 + (4.0 / x)));
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -4.4e-72:
		tmp = -1.0 / math.sqrt(((x + 1.0) / (-1.0 + x)))
	elif t <= -1.65e-109:
		tmp = (t / l) * math.sqrt(x)
	elif t <= -6.5e-184:
		tmp = -1.0
	elif t <= 1.06e-197:
		tmp = (t / l) / math.pow(x, -0.5)
	else:
		tmp = math.sqrt(2.0) / math.sqrt((2.0 + (4.0 / x)))
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -4.4e-72)
		tmp = Float64(-1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))));
	elseif (t <= -1.65e-109)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= -6.5e-184)
		tmp = -1.0;
	elseif (t <= 1.06e-197)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	else
		tmp = Float64(sqrt(2.0) / sqrt(Float64(2.0 + Float64(4.0 / x))));
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4.4e-72)
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	elseif (t <= -1.65e-109)
		tmp = (t / l) * sqrt(x);
	elseif (t <= -6.5e-184)
		tmp = -1.0;
	elseif (t <= 1.06e-197)
		tmp = (t / l) / (x ^ -0.5);
	else
		tmp = sqrt(2.0) / sqrt((2.0 + (4.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -4.4e-72], N[(-1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.65e-109], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.5e-184], -1.0, If[LessEqual[t, 1.06e-197], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[N[(2.0 + N[(4.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.40000000000000005e-72

   1. Initial program 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in t around -inf 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. +-commutative 84.4%

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

      4. sub-neg 84.4%

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

      5. metadata-eval 84.4%

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

      6. +-commutative 84.4%

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

   6. Simplified 84.4%

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

      1. frac-2neg 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

      4. distribute-lft-neg-in 84.4%

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

      5. times-frac 84.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{-\left(-\sqrt{2}\right)} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}} \]

      6. neg-mul-1 84.4%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{-1 \cdot \sqrt{2}}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      7. *-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{2} \cdot -1}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      8. distribute-rgt-neg-in 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(--1\right)}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      9. metadata-eval 84.4%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \color{blue}{1}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      10. +-commutative 84.4%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot 1} \cdot \frac{-1}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}}}} \]

   8. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2} \cdot 1} \cdot \frac{-1}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.4%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot 1} \cdot -1}{\sqrt{\frac{x + 1}{x + -1}}}} \]

      2. *-rgt-identity 84.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{\color{blue}{\sqrt{2}}} \cdot -1}{\sqrt{\frac{x + 1}{x + -1}}} \]

      3. *-inverses 84.4%

         \[\leadsto \frac{\color{blue}{1} \cdot -1}{\sqrt{\frac{x + 1}{x + -1}}} \]

      4. metadata-eval 84.4%

         \[\leadsto \frac{\color{blue}{-1}}{\sqrt{\frac{x + 1}{x + -1}}} \]

   10. Simplified 84.4%

       \[\leadsto \color{blue}{\frac{-1}{\sqrt{\frac{x + 1}{x + -1}}}} \]

   if -4.40000000000000005e-72 < t < -1.64999999999999995e-109

   1. Initial program 17.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 17.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 0.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 49.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]

   6. Taylor expanded in l around 0 51.1%

      \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]

   if -1.64999999999999995e-109 < t < -6.4999999999999997e-184

   1. Initial program 31.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{-1} \]

   if -6.4999999999999997e-184 < t < 1.05999999999999997e-197

   1. Initial program 3.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 1.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 39.3%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]
   6. Step-by-step derivation

      1. associate-/r* 39.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{\ell}{t}}}{\sqrt{\frac{2}{x}}}} \]

      2. div-inv 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\ell}{t}} \cdot \frac{1}{\sqrt{\frac{2}{x}}}} \]

      3. times-frac 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot 1}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}}} \]

      4. *-commutative 39.3%

         \[\leadsto \frac{\sqrt{2} \cdot 1}{\color{blue}{\sqrt{\frac{2}{x}} \cdot \frac{\ell}{t}}} \]

      5. times-frac 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{2}{x}}} \cdot \frac{1}{\frac{\ell}{t}}} \]

      6. clear-num 39.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{2}{x}}} \cdot \color{blue}{\frac{t}{\ell}} \]

      7. times-frac 44.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}} \]

      8. *-commutative 44.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x}}}} \]

      9. sqrt-div 44.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}} \]

      10. associate-*r/ 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}}} \]

      11. un-div-inv 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}} \]

      12. metadata-eval 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}} \]

      13. sqrt-div 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      14. *-commutative 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(\ell \cdot \sqrt{2}\right)}} \]

      15. times-frac 39.3%

          \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{1}{x}}} \cdot \frac{t}{\ell \cdot \sqrt{2}}} \]

      16. inv-pow 39.3%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{{x}^{-1}}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

      17. sqrt-pow1 39.3%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

      18. metadata-eval 39.3%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 39.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell \cdot \sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-/r* 39.3%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 39.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 39.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 39.3%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 44.1%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 44.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 44.1%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 39.3%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 39.3%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 1.05999999999999997e-197 < t

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 50.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}}{t}} \]
   5. Step-by-step derivation

      1. fma-def 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      2. unpow2 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      3. fma-def 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \color{blue}{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      4. unpow2 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \color{blue}{t \cdot t}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      5. unpow2 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      6. associate-*r/ 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \color{blue}{\frac{-1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{x}}}}{t}} \]

      7. mul-1-neg 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{\color{blue}{-\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}}}{t}} \]

      8. unpow2 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}}}{t}} \]

      9. fma-def 50.6%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}}}{t}} \]

      10. unpow2 50.6%

          \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}}}{t}} \]

   6. Simplified 50.6%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}}{t}} \]

   7. Taylor expanded in t around inf 82.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{t \cdot \sqrt{2 + 4 \cdot \frac{1}{x}}}}{t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 82.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{t \cdot \sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}}{t}} \]

      2. metadata-eval 82.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{t \cdot \sqrt{2 + \frac{\color{blue}{4}}{x}}}{t}} \]

   9. Simplified 82.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{t \cdot \sqrt{2 + \frac{4}{x}}}}{t}} \]

   10. Taylor expanded in t around 0 82.5%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + 4 \cdot \frac{1}{x}}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 82.5%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2 + \color{blue}{\frac{4 \cdot 1}{x}}}} \]

       2. metadata-eval 82.5%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2 + \frac{\color{blue}{4}}{x}}} \]

   12. Simplified 82.5%

       \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2 + \frac{4}{x}}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -1.65 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-184}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{\sqrt{2 + \frac{4}{x}}}\\ \end{array} \]

Alternative 11: 76.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-183}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -4.4e-72)
   (- (+ -1.0 (/ 1.0 x)) (/ 0.5 (* x x)))
   (if (<= t -3.6e-109)
     (* (/ t l) (sqrt x))
     (if (<= t -1.7e-183)
       -1.0
       (if (<= t 6.2e-199) (/ (/ t l) (pow x -0.5)) 1.0)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.4e-72) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else if (t <= -3.6e-109) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= -1.7e-183) {
		tmp = -1.0;
	} else if (t <= 6.2e-199) {
		tmp = (t / l) / pow(x, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.4d-72)) then
        tmp = ((-1.0d0) + (1.0d0 / x)) - (0.5d0 / (x * x))
    else if (t <= (-3.6d-109)) then
        tmp = (t / l) * sqrt(x)
    else if (t <= (-1.7d-183)) then
        tmp = -1.0d0
    else if (t <= 6.2d-199) then
        tmp = (t / l) / (x ** (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -4.4e-72) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else if (t <= -3.6e-109) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= -1.7e-183) {
		tmp = -1.0;
	} else if (t <= 6.2e-199) {
		tmp = (t / l) / Math.pow(x, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -4.4e-72:
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x))
	elif t <= -3.6e-109:
		tmp = (t / l) * math.sqrt(x)
	elif t <= -1.7e-183:
		tmp = -1.0
	elif t <= 6.2e-199:
		tmp = (t / l) / math.pow(x, -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -4.4e-72)
		tmp = Float64(Float64(-1.0 + Float64(1.0 / x)) - Float64(0.5 / Float64(x * x)));
	elseif (t <= -3.6e-109)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= -1.7e-183)
		tmp = -1.0;
	elseif (t <= 6.2e-199)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -4.4e-72)
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	elseif (t <= -3.6e-109)
		tmp = (t / l) * sqrt(x);
	elseif (t <= -1.7e-183)
		tmp = -1.0;
	elseif (t <= 6.2e-199)
		tmp = (t / l) / (x ^ -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -4.4e-72], N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.6e-109], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.7e-183], -1.0, If[LessEqual[t, 6.2e-199], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 5 regimes

2. if t < -4.40000000000000005e-72

   1. Initial program 36.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 84.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 84.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 83.4%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 83.4%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

      2. sub-neg 83.4%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

      3. metadata-eval 83.4%

         \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 83.4%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 83.4%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      6. unpow2 83.4%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

   9. Simplified 83.4%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

   if -4.40000000000000005e-72 < t < -3.6000000000000001e-109

   1. Initial program 17.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 17.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 0.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 49.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]

   6. Taylor expanded in l around 0 51.1%

      \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]

   if -3.6000000000000001e-109 < t < -1.70000000000000007e-183

   1. Initial program 31.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{-1} \]

   if -1.70000000000000007e-183 < t < 6.20000000000000024e-199

   1. Initial program 3.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 1.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 39.3%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]
   6. Step-by-step derivation

      1. associate-/r* 39.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{\ell}{t}}}{\sqrt{\frac{2}{x}}}} \]

      2. div-inv 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\ell}{t}} \cdot \frac{1}{\sqrt{\frac{2}{x}}}} \]

      3. times-frac 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot 1}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}}} \]

      4. *-commutative 39.3%

         \[\leadsto \frac{\sqrt{2} \cdot 1}{\color{blue}{\sqrt{\frac{2}{x}} \cdot \frac{\ell}{t}}} \]

      5. times-frac 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{2}{x}}} \cdot \frac{1}{\frac{\ell}{t}}} \]

      6. clear-num 39.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{2}{x}}} \cdot \color{blue}{\frac{t}{\ell}} \]

      7. times-frac 44.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}} \]

      8. *-commutative 44.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x}}}} \]

      9. sqrt-div 44.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}} \]

      10. associate-*r/ 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}}} \]

      11. un-div-inv 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}} \]

      12. metadata-eval 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}} \]

      13. sqrt-div 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      14. *-commutative 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(\ell \cdot \sqrt{2}\right)}} \]

      15. times-frac 39.3%

          \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{1}{x}}} \cdot \frac{t}{\ell \cdot \sqrt{2}}} \]

      16. inv-pow 39.3%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{{x}^{-1}}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

      17. sqrt-pow1 39.3%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

      18. metadata-eval 39.3%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 39.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell \cdot \sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-/r* 39.3%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 39.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 39.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 39.3%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 44.1%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 44.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 44.1%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 39.3%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 39.3%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 6.20000000000000024e-199 < t

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 80.7%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   5. Step-by-step derivation

      1. *-commutative 80.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   6. Simplified 80.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   7. Step-by-step derivation

      1. rewrite-binary64/binary32 45.7%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 45.7%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 81.9%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 81.9%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 81.9%

         \[\leadsto 1 \]

   10. Applied egg-rr 81.9%

       \[\leadsto 1 \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-72}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-183}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -6.5e-72)
   (/ -1.0 (sqrt (/ (+ x 1.0) (+ -1.0 x))))
   (if (<= t -7.8e-109)
     (* (/ t l) (sqrt x))
     (if (<= t -1.9e-185)
       -1.0
       (if (<= t 6e-197) (/ (/ t l) (pow x -0.5)) 1.0)))))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -6.5e-72) {
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -7.8e-109) {
		tmp = (t / l) * sqrt(x);
	} else if (t <= -1.9e-185) {
		tmp = -1.0;
	} else if (t <= 6e-197) {
		tmp = (t / l) / pow(x, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-6.5d-72)) then
        tmp = (-1.0d0) / sqrt(((x + 1.0d0) / ((-1.0d0) + x)))
    else if (t <= (-7.8d-109)) then
        tmp = (t / l) * sqrt(x)
    else if (t <= (-1.9d-185)) then
        tmp = -1.0d0
    else if (t <= 6d-197) then
        tmp = (t / l) / (x ** (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -6.5e-72) {
		tmp = -1.0 / Math.sqrt(((x + 1.0) / (-1.0 + x)));
	} else if (t <= -7.8e-109) {
		tmp = (t / l) * Math.sqrt(x);
	} else if (t <= -1.9e-185) {
		tmp = -1.0;
	} else if (t <= 6e-197) {
		tmp = (t / l) / Math.pow(x, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -6.5e-72:
		tmp = -1.0 / math.sqrt(((x + 1.0) / (-1.0 + x)))
	elif t <= -7.8e-109:
		tmp = (t / l) * math.sqrt(x)
	elif t <= -1.9e-185:
		tmp = -1.0
	elif t <= 6e-197:
		tmp = (t / l) / math.pow(x, -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -6.5e-72)
		tmp = Float64(-1.0 / sqrt(Float64(Float64(x + 1.0) / Float64(-1.0 + x))));
	elseif (t <= -7.8e-109)
		tmp = Float64(Float64(t / l) * sqrt(x));
	elseif (t <= -1.9e-185)
		tmp = -1.0;
	elseif (t <= 6e-197)
		tmp = Float64(Float64(t / l) / (x ^ -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -6.5e-72)
		tmp = -1.0 / sqrt(((x + 1.0) / (-1.0 + x)));
	elseif (t <= -7.8e-109)
		tmp = (t / l) * sqrt(x);
	elseif (t <= -1.9e-185)
		tmp = -1.0;
	elseif (t <= 6e-197)
		tmp = (t / l) / (x ^ -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -6.5e-72], N[(-1.0 / N[Sqrt[N[(N[(x + 1.0), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.8e-109], N[(N[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.9e-185], -1.0, If[LessEqual[t, 6e-197], N[(N[(t / l), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], 1.0]]]]

Derivation

1. Split input into 5 regimes

2. if t < -6.4999999999999997e-72

   1. Initial program 36.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 84.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 84.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]
   7. Step-by-step derivation

      1. frac-2neg 84.4%

         \[\leadsto \color{blue}{\frac{-\sqrt{2}}{-\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

      2. neg-mul-1 84.4%

         \[\leadsto \frac{\color{blue}{-1 \cdot \sqrt{2}}}{-\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      3. *-commutative 84.4%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot -1}}{-\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}} \]

      4. distribute-lft-neg-in 84.4%

         \[\leadsto \frac{\sqrt{2} \cdot -1}{\color{blue}{\left(-\left(-\sqrt{2}\right)\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

      5. times-frac 84.4%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{-\left(-\sqrt{2}\right)} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}} \]

      6. neg-mul-1 84.4%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{-1 \cdot \sqrt{2}}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      7. *-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{-\color{blue}{\sqrt{2} \cdot -1}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      8. distribute-rgt-neg-in 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\sqrt{2} \cdot \left(--1\right)}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      9. metadata-eval 84.4%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot \color{blue}{1}} \cdot \frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}} \]

      10. +-commutative 84.4%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{2} \cdot 1} \cdot \frac{-1}{\sqrt{\frac{x + 1}{\color{blue}{x + -1}}}} \]

   8. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2} \cdot 1} \cdot \frac{-1}{\sqrt{\frac{x + 1}{x + -1}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 84.4%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\sqrt{2} \cdot 1} \cdot -1}{\sqrt{\frac{x + 1}{x + -1}}}} \]

      2. *-rgt-identity 84.4%

         \[\leadsto \frac{\frac{\sqrt{2}}{\color{blue}{\sqrt{2}}} \cdot -1}{\sqrt{\frac{x + 1}{x + -1}}} \]

      3. *-inverses 84.4%

         \[\leadsto \frac{\color{blue}{1} \cdot -1}{\sqrt{\frac{x + 1}{x + -1}}} \]

      4. metadata-eval 84.4%

         \[\leadsto \frac{\color{blue}{-1}}{\sqrt{\frac{x + 1}{x + -1}}} \]

   10. Simplified 84.4%

       \[\leadsto \color{blue}{\frac{-1}{\sqrt{\frac{x + 1}{x + -1}}}} \]

   if -6.4999999999999997e-72 < t < -7.80000000000000046e-109

   1. Initial program 17.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 17.2%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 0.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 49.0%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]

   6. Taylor expanded in l around 0 51.1%

      \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]

   if -7.80000000000000046e-109 < t < -1.9e-185

   1. Initial program 31.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{-1} \]

   if -1.9e-185 < t < 6.00000000000000051e-197

   1. Initial program 3.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 1.0%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 39.3%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]
   6. Step-by-step derivation

      1. associate-/r* 39.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{\frac{\ell}{t}}}{\sqrt{\frac{2}{x}}}} \]

      2. div-inv 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\ell}{t}} \cdot \frac{1}{\sqrt{\frac{2}{x}}}} \]

      3. times-frac 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot 1}{\frac{\ell}{t} \cdot \sqrt{\frac{2}{x}}}} \]

      4. *-commutative 39.3%

         \[\leadsto \frac{\sqrt{2} \cdot 1}{\color{blue}{\sqrt{\frac{2}{x}} \cdot \frac{\ell}{t}}} \]

      5. times-frac 39.3%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{2}{x}}} \cdot \frac{1}{\frac{\ell}{t}}} \]

      6. clear-num 39.3%

         \[\leadsto \frac{\sqrt{2}}{\sqrt{\frac{2}{x}}} \cdot \color{blue}{\frac{t}{\ell}} \]

      7. times-frac 44.0%

         \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{\frac{2}{x}} \cdot \ell}} \]

      8. *-commutative 44.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\ell \cdot \sqrt{\frac{2}{x}}}} \]

      9. sqrt-div 44.0%

         \[\leadsto \frac{\sqrt{2} \cdot t}{\ell \cdot \color{blue}{\frac{\sqrt{2}}{\sqrt{x}}}} \]

      10. associate-*r/ 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\frac{\ell \cdot \sqrt{2}}{\sqrt{x}}}} \]

      11. un-div-inv 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{1}{\sqrt{x}}}} \]

      12. metadata-eval 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\color{blue}{\sqrt{1}}}{\sqrt{x}}} \]

      13. sqrt-div 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\left(\ell \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{1}{x}}}} \]

      14. *-commutative 44.1%

          \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(\ell \cdot \sqrt{2}\right)}} \]

      15. times-frac 39.3%

          \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{\frac{1}{x}}} \cdot \frac{t}{\ell \cdot \sqrt{2}}} \]

      16. inv-pow 39.3%

          \[\leadsto \frac{\sqrt{2}}{\sqrt{\color{blue}{{x}^{-1}}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

      17. sqrt-pow1 39.3%

          \[\leadsto \frac{\sqrt{2}}{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

      18. metadata-eval 39.3%

          \[\leadsto \frac{\sqrt{2}}{{x}^{\color{blue}{-0.5}}} \cdot \frac{t}{\ell \cdot \sqrt{2}} \]

   7. Applied egg-rr 39.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell \cdot \sqrt{2}}} \]
   8. Step-by-step derivation

      1. associate-/r* 39.3%

         \[\leadsto \frac{\sqrt{2}}{{x}^{-0.5}} \cdot \color{blue}{\frac{\frac{t}{\ell}}{\sqrt{2}}} \]

      2. associate-*r/ 39.3%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{{x}^{-0.5}} \cdot \frac{t}{\ell}}{\sqrt{2}}} \]

      3. associate-*l/ 39.3%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt{2} \cdot \frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      4. associate-*r/ 39.3%

         \[\leadsto \frac{\color{blue}{\sqrt{2} \cdot \frac{\frac{t}{\ell}}{{x}^{-0.5}}}}{\sqrt{2}} \]

      5. associate-/r* 44.1%

         \[\leadsto \frac{\sqrt{2} \cdot \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}}}{\sqrt{2}} \]

      6. associate-*l/ 44.1%

         \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

      7. *-inverses 44.1%

         \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

      8. associate-/r* 39.3%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

      9. *-lft-identity 39.3%

         \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   9. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{\frac{t}{\ell}}{{x}^{-0.5}}} \]

   if 6.00000000000000051e-197 < t

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 80.7%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   5. Step-by-step derivation

      1. *-commutative 80.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   6. Simplified 80.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   7. Step-by-step derivation

      1. rewrite-binary64/binary32 45.7%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 45.7%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 81.9%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 81.9%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 81.9%

         \[\leadsto 1 \]

   10. Applied egg-rr 81.9%

       \[\leadsto 1 \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{-1}{\sqrt{\frac{x + 1}{-1 + x}}}\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -1.9 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-197}:\\ \;\;\;\;\frac{\frac{t}{\ell}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 76.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} t_1 := \frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (let* ((t_1 (* (/ t l) (sqrt x))))
   (if (<= t -8.8e-72)
     (- (+ -1.0 (/ 1.0 x)) (/ 0.5 (* x x)))
     (if (<= t -1.3e-114)
       t_1
       (if (<= t -2.75e-185) -1.0 (if (<= t 5.2e-198) t_1 1.0))))))

C

l = abs(l);
double code(double x, double l, double t) {
	double t_1 = (t / l) * sqrt(x);
	double tmp;
	if (t <= -8.8e-72) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else if (t <= -1.3e-114) {
		tmp = t_1;
	} else if (t <= -2.75e-185) {
		tmp = -1.0;
	} else if (t <= 5.2e-198) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / l) * sqrt(x)
    if (t <= (-8.8d-72)) then
        tmp = ((-1.0d0) + (1.0d0 / x)) - (0.5d0 / (x * x))
    else if (t <= (-1.3d-114)) then
        tmp = t_1
    else if (t <= (-2.75d-185)) then
        tmp = -1.0d0
    else if (t <= 5.2d-198) then
        tmp = t_1
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double t_1 = (t / l) * Math.sqrt(x);
	double tmp;
	if (t <= -8.8e-72) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else if (t <= -1.3e-114) {
		tmp = t_1;
	} else if (t <= -2.75e-185) {
		tmp = -1.0;
	} else if (t <= 5.2e-198) {
		tmp = t_1;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	t_1 = (t / l) * math.sqrt(x)
	tmp = 0
	if t <= -8.8e-72:
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x))
	elif t <= -1.3e-114:
		tmp = t_1
	elif t <= -2.75e-185:
		tmp = -1.0
	elif t <= 5.2e-198:
		tmp = t_1
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	t_1 = Float64(Float64(t / l) * sqrt(x))
	tmp = 0.0
	if (t <= -8.8e-72)
		tmp = Float64(Float64(-1.0 + Float64(1.0 / x)) - Float64(0.5 / Float64(x * x)));
	elseif (t <= -1.3e-114)
		tmp = t_1;
	elseif (t <= -2.75e-185)
		tmp = -1.0;
	elseif (t <= 5.2e-198)
		tmp = t_1;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	t_1 = (t / l) * sqrt(x);
	tmp = 0.0;
	if (t <= -8.8e-72)
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	elseif (t <= -1.3e-114)
		tmp = t_1;
	elseif (t <= -2.75e-185)
		tmp = -1.0;
	elseif (t <= 5.2e-198)
		tmp = t_1;
	else
		tmp = 1.0;
	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[(t / l), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.8e-72], N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.3e-114], t$95$1, If[LessEqual[t, -2.75e-185], -1.0, If[LessEqual[t, 5.2e-198], t$95$1, 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.8000000000000001e-72

   1. Initial program 36.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 84.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 84.4%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 84.4%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 84.4%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 83.4%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 83.4%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

      2. sub-neg 83.4%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

      3. metadata-eval 83.4%

         \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 83.4%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 83.4%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      6. unpow2 83.4%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

   9. Simplified 83.4%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

   if -8.8000000000000001e-72 < t < -1.30000000000000007e-114 or -2.7499999999999999e-185 < t < 5.20000000000000014e-198

   1. Initial program 7.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 7.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in l around inf 0.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(\frac{1}{x - 1} + \frac{x}{x - 1}\right) - 1}}} \]

   5. Taylor expanded in x around inf 41.9%

      \[\leadsto \frac{\sqrt{2}}{\frac{\ell}{t} \cdot \sqrt{\color{blue}{\frac{2}{x}}}} \]

   6. Taylor expanded in l around 0 42.5%

      \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]

   if -1.30000000000000007e-114 < t < -2.7499999999999999e-185

   1. Initial program 31.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 31.5%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 93.6%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 93.6%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 93.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 93.6%

      \[\leadsto \color{blue}{-1} \]

   if 5.20000000000000014e-198 < t

   1. Initial program 40.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 80.7%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   5. Step-by-step derivation

      1. *-commutative 80.7%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   6. Simplified 80.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   7. Step-by-step derivation

      1. rewrite-binary64/binary32 45.7%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 45.7%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 81.9%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 81.9%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 81.9%

         \[\leadsto 1 \]

   10. Applied egg-rr 81.9%

       \[\leadsto 1 \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-72}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq -2.75 \cdot 10^{-185}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 75.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.42 \cdot 10^{-244}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{0}{{2}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.42e-244)
   (- (+ -1.0 (/ 1.0 x)) (/ 0.5 (* x x)))
   (if (<= t 2.8e-252) (/ 0.0 (pow 2.0 -0.5)) 1.0)))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.42e-244) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else if (t <= 2.8e-252) {
		tmp = 0.0 / pow(2.0, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.42d-244)) then
        tmp = ((-1.0d0) + (1.0d0 / x)) - (0.5d0 / (x * x))
    else if (t <= 2.8d-252) then
        tmp = 0.0d0 / (2.0d0 ** (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.42e-244) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else if (t <= 2.8e-252) {
		tmp = 0.0 / Math.pow(2.0, -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1.42e-244:
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x))
	elif t <= 2.8e-252:
		tmp = 0.0 / math.pow(2.0, -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1.42e-244)
		tmp = Float64(Float64(-1.0 + Float64(1.0 / x)) - Float64(0.5 / Float64(x * x)));
	elseif (t <= 2.8e-252)
		tmp = Float64(0.0 / (2.0 ^ -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1.42e-244)
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	elseif (t <= 2.8e-252)
		tmp = 0.0 / (2.0 ^ -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1.42e-244], N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e-252], N[(0.0 / N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if t < -1.42000000000000003e-244

   1. Initial program 31.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 74.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 74.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 74.9%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 74.9%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 74.9%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 74.9%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 74.9%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 74.9%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 74.2%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 74.2%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

      2. sub-neg 74.2%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

      3. metadata-eval 74.2%

         \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 74.2%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 74.2%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      6. unpow2 74.2%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

   9. Simplified 74.2%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

   if -1.42000000000000003e-244 < t < 2.80000000000000018e-252

   1. Initial program 1.4%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 1.4%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 63.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}}{t}} \]
   5. Step-by-step derivation

      1. fma-def 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)} - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      2. unpow2 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{x}, 2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      3. fma-def 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \color{blue}{\mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)}\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      4. unpow2 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, \color{blue}{t \cdot t}, \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      5. unpow2 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\color{blue}{\ell \cdot \ell}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}{t}} \]

      6. associate-*r/ 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \color{blue}{\frac{-1 \cdot \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{x}}}}{t}} \]

      7. mul-1-neg 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{\color{blue}{-\left(2 \cdot {t}^{2} + {\ell}^{2}\right)}}{x}}}{t}} \]

      8. unpow2 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\left(2 \cdot {t}^{2} + \color{blue}{\ell \cdot \ell}\right)}{x}}}{t}} \]

      9. fma-def 63.4%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\color{blue}{\mathsf{fma}\left(2, {t}^{2}, \ell \cdot \ell\right)}}{x}}}{t}} \]

      10. unpow2 63.4%

          \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(2, \color{blue}{t \cdot t}, \ell \cdot \ell\right)}{x}}}{t}} \]

   6. Simplified 63.4%

      \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\mathsf{fma}\left(2, \frac{t \cdot t}{x}, \mathsf{fma}\left(2, t \cdot t, \frac{\ell \cdot \ell}{x}\right)\right) - \frac{-\mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right)}{x}}}}{t}} \]

   7. Taylor expanded in t around 0 63.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{\sqrt{\frac{{\ell}^{2}}{x} - -1 \cdot \frac{{\ell}^{2}}{x}}}}{t}} \]
   8. Step-by-step derivation

      1. cancel-sign-sub-inv 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\frac{{\ell}^{2}}{x} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}}}{t}} \]

      2. unpow2 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\frac{\color{blue}{\ell \cdot \ell}}{x} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}}{t}} \]

      3. associate-*r/ 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\color{blue}{\ell \cdot \frac{\ell}{x}} + \left(--1\right) \cdot \frac{{\ell}^{2}}{x}}}{t}} \]

      4. metadata-eval 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \color{blue}{1} \cdot \frac{{\ell}^{2}}{x}}}{t}} \]

      5. associate-*r/ 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \color{blue}{\frac{1 \cdot {\ell}^{2}}{x}}}}{t}} \]

      6. *-commutative 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \frac{\color{blue}{{\ell}^{2} \cdot 1}}{x}}}{t}} \]

      7. *-rgt-identity 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \frac{\color{blue}{{\ell}^{2}}}{x}}}{t}} \]

      8. unpow2 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \frac{\color{blue}{\ell \cdot \ell}}{x}}}{t}} \]

      9. associate-*r/ 63.1%

         \[\leadsto \frac{\sqrt{2}}{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \color{blue}{\ell \cdot \frac{\ell}{x}}}}{t}} \]

   9. Simplified 63.1%

      \[\leadsto \frac{\sqrt{2}}{\frac{\color{blue}{\sqrt{\ell \cdot \frac{\ell}{x} + \ell \cdot \frac{\ell}{x}}}}{t}} \]
   10. Step-by-step derivation

       1. clear-num 63.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \ell \cdot \frac{\ell}{x}}}{t}}{\sqrt{2}}}} \]

       2. inv-pow 63.1%

          \[\leadsto \color{blue}{{\left(\frac{\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \ell \cdot \frac{\ell}{x}}}{t}}{\sqrt{2}}\right)}^{-1}} \]

       3. div-inv 63.1%

          \[\leadsto {\color{blue}{\left(\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \ell \cdot \frac{\ell}{x}}}{t} \cdot \frac{1}{\sqrt{2}}\right)}}^{-1} \]

       4. unpow-prod-down 63.1%

          \[\leadsto \color{blue}{{\left(\frac{\sqrt{\ell \cdot \frac{\ell}{x} + \ell \cdot \frac{\ell}{x}}}{t}\right)}^{-1} \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{-1}} \]

   11. Applied egg-rr 58.6%

       \[\leadsto \color{blue}{\left(t \cdot 0\right) \cdot {\left({2}^{-0.5}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. mul0-rgt 58.6%

          \[\leadsto \color{blue}{0} \cdot {\left({2}^{-0.5}\right)}^{-1} \]

       2. unpow-1 58.6%

          \[\leadsto 0 \cdot \color{blue}{\frac{1}{{2}^{-0.5}}} \]

       3. associate-*r/ 58.6%

          \[\leadsto \color{blue}{\frac{0 \cdot 1}{{2}^{-0.5}}} \]

       4. metadata-eval 58.6%

          \[\leadsto \frac{\color{blue}{0}}{{2}^{-0.5}} \]

   13. Simplified 58.6%

       \[\leadsto \color{blue}{\frac{0}{{2}^{-0.5}}} \]

   if 2.80000000000000018e-252 < t

   1. Initial program 38.7%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 38.7%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 78.3%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   5. Step-by-step derivation

      1. *-commutative 78.3%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   6. Simplified 78.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   7. Step-by-step derivation

      1. rewrite-binary64/binary32 44.4%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 44.4%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 79.5%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 79.5%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 79.5%

         \[\leadsto 1 \]

   10. Applied egg-rr 79.5%

       \[\leadsto 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.42 \cdot 10^{-244}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{0}{{2}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 75.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t)
 :precision binary64
 (if (<= t -1e-309) (- (+ -1.0 (/ 1.0 x)) (/ 0.5 (* x x))) 1.0))

C

l = abs(l);
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: l should be positive before calling this function
real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1d-309)) then
        tmp = ((-1.0d0) + (1.0d0 / x)) - (0.5d0 / (x * x))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	double tmp;
	if (t <= -1e-309) {
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

l = abs(l)
def code(x, l, t):
	tmp = 0
	if t <= -1e-309:
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x))
	else:
		tmp = 1.0
	return tmp

Julia

l = abs(l)
function code(x, l, t)
	tmp = 0.0
	if (t <= -1e-309)
		tmp = Float64(Float64(-1.0 + Float64(1.0 / x)) - Float64(0.5 / Float64(x * x)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

l = abs(l)
function tmp_2 = code(x, l, t)
	tmp = 0.0;
	if (t <= -1e-309)
		tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := If[LessEqual[t, -1e-309], N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if t < -1.000000000000002e-309

   1. Initial program 29.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 29.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in t around -inf 72.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r* 72.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

      2. neg-mul-1 72.5%

         \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

      3. +-commutative 72.5%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      4. sub-neg 72.5%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      5. metadata-eval 72.5%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      6. +-commutative 72.5%

         \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   6. Simplified 72.5%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   7. Taylor expanded in x around inf 71.8%

      \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
   8. Step-by-step derivation

      1. associate--r+ 71.8%

         \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

      2. sub-neg 71.8%

         \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

      3. metadata-eval 71.8%

         \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 71.8%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 71.8%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

      6. unpow2 71.8%

         \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

   9. Simplified 71.8%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

   if -1.000000000000002e-309 < t

   1. Initial program 36.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 36.8%

      \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

   4. Taylor expanded in x around inf 75.3%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \sqrt{2}} \]
   5. Step-by-step derivation

      1. *-commutative 75.3%

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]

   6. Simplified 75.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \]
   7. Step-by-step derivation

      1. rewrite-binary64/binary32 42.8%

         \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]

   8. Applied rewrite-once 42.8%

      \[\leadsto \color{blue}{\langle \color{blue}{\left( \color{blue}{\sqrt{2} \cdot \sqrt{0.5}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} \]
   9. Step-by-step derivation

      1. sqrt-unprod 76.5%

         \[\leadsto \langle \left( \sqrt{\color{blue}{2 \cdot 0.5}} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      2. metadata-eval 76.5%

         \[\leadsto \langle \left( \sqrt{1} \right)_{\text{binary32}} \rangle_{\text{binary64}} \]

      3. metadata-eval 76.5%

         \[\leadsto 1 \]

   10. Applied egg-rr 76.5%

       \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 38.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} l = |l|\\ \\ \left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x} \end{array} \]

FPCore

NOTE: l should be positive before calling this function
(FPCore (x l t) :precision binary64 (- (+ -1.0 (/ 1.0 x)) (/ 0.5 (* x x))))

C

l = abs(l);
double code(double x, double l, double t) {
	return (-1.0 + (1.0 / x)) - (0.5 / (x * 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)) - (0.5d0 / (x * x))
end function

Java

l = Math.abs(l);
public static double code(double x, double l, double t) {
	return (-1.0 + (1.0 / x)) - (0.5 / (x * x));
}

Python

l = abs(l)
def code(x, l, t):
	return (-1.0 + (1.0 / x)) - (0.5 / (x * x))

Julia

l = abs(l)
function code(x, l, t)
	return Float64(Float64(-1.0 + Float64(1.0 / x)) - Float64(0.5 / Float64(x * x)))
end

MATLAB

l = abs(l)
function tmp = code(x, l, t)
	tmp = (-1.0 + (1.0 / x)) - (0.5 / (x * x));
end

Wolfram

NOTE: l should be positive before calling this function
code[x_, l_, t_] := N[(N[(-1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

3. Simplified 33.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

4. Taylor expanded in t around -inf 39.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 39.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. neg-mul-1 39.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   3. +-commutative 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

   4. sub-neg 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

   5. metadata-eval 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   6. +-commutative 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

6. Simplified 39.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

7. Taylor expanded in x around inf 39.2%

   \[\leadsto \color{blue}{\frac{1}{x} - \left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right)} \]
8. Step-by-step derivation

   1. associate--r+ 39.2%

      \[\leadsto \color{blue}{\left(\frac{1}{x} - 1\right) - 0.5 \cdot \frac{1}{{x}^{2}}} \]

   2. sub-neg 39.2%

      \[\leadsto \color{blue}{\left(\frac{1}{x} + \left(-1\right)\right)} - 0.5 \cdot \frac{1}{{x}^{2}} \]

   3. metadata-eval 39.2%

      \[\leadsto \left(\frac{1}{x} + \color{blue}{-1}\right) - 0.5 \cdot \frac{1}{{x}^{2}} \]

   4. associate-*r/ 39.2%

      \[\leadsto \left(\frac{1}{x} + -1\right) - \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} \]

   5. metadata-eval 39.2%

      \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{\color{blue}{0.5}}{{x}^{2}} \]

   6. unpow2 39.2%

      \[\leadsto \left(\frac{1}{x} + -1\right) - \frac{0.5}{\color{blue}{x \cdot x}} \]

9. Simplified 39.2%

   \[\leadsto \color{blue}{\left(\frac{1}{x} + -1\right) - \frac{0.5}{x \cdot x}} \]

10. Final simplification 39.2%

    \[\leadsto \left(-1 + \frac{1}{x}\right) - \frac{0.5}{x \cdot x} \]

Alternative 17: 38.8% 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 33.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

3. Simplified 33.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

4. Taylor expanded in t around -inf 39.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 39.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. neg-mul-1 39.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   3. +-commutative 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

   4. sub-neg 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

   5. metadata-eval 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   6. +-commutative 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

6. Simplified 39.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

7. Taylor expanded in x around inf 39.0%

   \[\leadsto \color{blue}{\frac{1}{x} - 1} \]

8. Final simplification 39.0%

   \[\leadsto -1 + \frac{1}{x} \]

Alternative 18: 38.5% 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 33.0%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

3. Simplified 33.0%

   \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}{t}}} \]

4. Taylor expanded in t around -inf 39.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{-1 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{1 + x}{x - 1}}\right)}} \]
5. Step-by-step derivation

   1. associate-*r* 39.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-1 \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]

   2. neg-mul-1 39.6%

      \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{1 + x}{x - 1}}} \]

   3. +-commutative 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

   4. sub-neg 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

   5. metadata-eval 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   6. +-commutative 39.6%

      \[\leadsto \frac{\sqrt{2}}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

6. Simplified 39.6%

   \[\leadsto \frac{\sqrt{2}}{\color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

7. Taylor expanded in x around inf 38.3%

   \[\leadsto \color{blue}{-1} \]

8. Final simplification 38.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023297 
(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)))))