Toniolo and Linder, Equation (7)

Percentage Accurate: 33.5% → 84.2%

Time: 40.8s

Alternatives: 14

Speedup: 225.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 33.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t_4 := t\_3 + t\_3\\ t_5 := \mathsf{fma}\left(l\_m, l\_m, {l\_m}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.26 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_4}{t\_m \cdot \left(x \cdot \sqrt{2}\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 1.18 \cdot 10^{-163}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{t\_5} + \frac{x}{t\_5}}\right)\\ \mathbf{elif}\;t\_m \leq 10^{+47}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(\frac{t\_4}{{x}^{2}} + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right)\right) + \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l_m 2.0)))
        (t_4 (+ t_3 t_3))
        (t_5 (fma l_m l_m (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.6e-227)
      (/ (/ t_m (pow x -0.5)) l_m)
      (if (<= t_m 1.26e-188)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+ (* 0.5 (/ t_4 (* t_m (* x (sqrt 2.0))))) (* t_m (sqrt 2.0)))))
        (if (<= t_m 1.18e-163)
          (* t_m (* (sqrt 2.0) (sqrt (+ (/ -1.0 t_5) (/ x t_5)))))
          (if (<= t_m 1e+47)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (+
                 (/ t_4 (pow x 2.0))
                 (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x))))
                (/ t_3 x)))))
            (+
             1.0
             (+
              (/ 0.5 (pow x 2.0))
              (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double t_4 = t_3 + t_3;
	double t_5 = fma(l_m, l_m, pow(l_m, 2.0));
	double tmp;
	if (t_m <= 1.6e-227) {
		tmp = (t_m / pow(x, -0.5)) / l_m;
	} else if (t_m <= 1.26e-188) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (t_4 / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.18e-163) {
		tmp = t_m * (sqrt(2.0) * sqrt(((-1.0 / t_5) + (x / t_5))));
	} else if (t_m <= 1e+47) {
		tmp = sqrt(2.0) * (t_m / sqrt((((t_4 / pow(x, 2.0)) + ((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x)))) + (t_3 / x))));
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	t_4 = Float64(t_3 + t_3)
	t_5 = fma(l_m, l_m, (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.6e-227)
		tmp = Float64(Float64(t_m / (x ^ -0.5)) / l_m);
	elseif (t_m <= 1.26e-188)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(t_4 / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.18e-163)
		tmp = Float64(t_m * Float64(sqrt(2.0) * sqrt(Float64(Float64(-1.0 / t_5) + Float64(x / t_5)))));
	elseif (t_m <= 1e+47)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(t_4 / (x ^ 2.0)) + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x)))) + Float64(t_3 / x)))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(l$95$m * l$95$m + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.6e-227], N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.26e-188], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(t$95$4 / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.18e-163], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / t$95$5), $MachinePrecision] + N[(x / t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1e+47], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(t$95$4 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < 1.60000000000000005e-227

   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%

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

   5. Taylor expanded in x around inf 58.2%

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

   6. Taylor expanded in l around inf 20.8%

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

      1. *-commutative 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. pow1/2 20.8%

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

      3. inv-pow 20.8%

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

      4. pow-pow 20.8%

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

      5. metadata-eval 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

   12. Simplified 20.8%

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

       1. associate-*r* 20.8%

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

       2. associate-*r/ 20.8%

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

       3. times-frac 20.9%

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

       4. sqrt-undiv 20.9%

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

       5. metadata-eval 20.9%

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

       6. metadata-eval 20.9%

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

       7. metadata-eval 20.9%

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

       8. times-frac 20.9%

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

       9. *-un-lft-identity 20.9%

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

       10. *-un-lft-identity 20.9%

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

   14. Applied egg-rr 20.9%

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

       1. *-lft-identity 20.9%

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

       2. associate-*r/ 20.9%

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

       3. times-frac 20.9%

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

       4. associate-*l/ 20.9%

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

       5. *-lft-identity 20.9%

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

   16. Simplified 20.9%

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

   if 1.60000000000000005e-227 < t < 1.26e-188

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in x around inf 68.3%

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

   if 1.26e-188 < t < 1.18000000000000004e-163

   1. Initial program 2.4%

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

   3. Simplified 3.4%

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

      1. clear-num 5.4%

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

      2. inv-pow 5.4%

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

      3. pow2 5.4%

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

      4. pow2 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. unpow-1 5.4%

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

      2. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in t around 0 6.6%

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

       1. associate-*l* 6.6%

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

       2. associate-/l* 2.4%

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

       3. +-commutative 2.4%

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

       4. sub-neg 2.4%

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

       5. metadata-eval 2.4%

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

       6. fma-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in x around 0 99.0%

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

       1. +-commutative 99.0%

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

       2. sub-neg 99.0%

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

       3. neg-mul-1 99.0%

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

       4. distribute-neg-out 99.0%

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

       5. *-lft-identity 99.0%

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

       6. metadata-eval 99.0%

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

       7. cancel-sign-sub-inv 99.0%

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

       8. distribute-neg-frac2 99.0%

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

       9. distribute-neg-frac 99.0%

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

       10. metadata-eval 99.0%

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

       11. unpow2 99.0%

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

       12. fma-neg 99.0%

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

       13. neg-mul-1 99.0%

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

       14. remove-double-neg 99.0%

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

       15. mul-1-neg 99.0%

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

       16. distribute-neg-frac2 99.0%

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

   14. Simplified 99.0%

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

   if 1.18000000000000004e-163 < t < 1e47

   1. Initial program 51.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around -inf 85.8%

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

   if 1e47 < t

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in t around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 97.7%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 97.7%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 97.7%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 97.7%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 97.7%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)} + \frac{x}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)}}\right)\\ \mathbf{elif}\;t \leq 10^{+47}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(\frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{{x}^{2}} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := 2 \cdot {t\_m}^{2} + {l\_m}^{2}\\ t_4 := \mathsf{fma}\left(l\_m, l\_m, {l\_m}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.25 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(x \cdot \sqrt{2}\right)} + t\_2}\\ \mathbf{elif}\;t\_m \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{t\_4} + \frac{x}{t\_4}}\right)\\ \mathbf{elif}\;t\_m \leq 3.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{t\_2}{\sqrt{\mathsf{fma}\left(2, \frac{{t\_m}^{2}}{x}, \mathsf{fma}\left(2, {t\_m}^{2}, \frac{{l\_m}^{2}}{x}\right)\right) + \frac{\mathsf{fma}\left(2, {t\_m}^{2}, {l\_m}^{2}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3 (+ (* 2.0 (pow t_m 2.0)) (pow l_m 2.0)))
        (t_4 (fma l_m l_m (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 1.8e-226)
      (/ (/ t_m (pow x -0.5)) l_m)
      (if (<= t_m 1.25e-188)
        (*
         (sqrt 2.0)
         (/ t_m (+ (* 0.5 (/ (+ t_3 t_3) (* t_m (* x (sqrt 2.0))))) t_2)))
        (if (<= t_m 7.2e-163)
          (* t_m (* (sqrt 2.0) (sqrt (+ (/ -1.0 t_4) (/ x t_4)))))
          (if (<= t_m 3.4e+47)
            (/
             t_2
             (sqrt
              (+
               (fma
                2.0
                (/ (pow t_m 2.0) x)
                (fma 2.0 (pow t_m 2.0) (/ (pow l_m 2.0) x)))
               (/ (fma 2.0 (pow t_m 2.0) (pow l_m 2.0)) x))))
            (+
             1.0
             (+
              (/ 0.5 (pow x 2.0))
              (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = (2.0 * pow(t_m, 2.0)) + pow(l_m, 2.0);
	double t_4 = fma(l_m, l_m, pow(l_m, 2.0));
	double tmp;
	if (t_m <= 1.8e-226) {
		tmp = (t_m / pow(x, -0.5)) / l_m;
	} else if (t_m <= 1.25e-188) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0))))) + t_2));
	} else if (t_m <= 7.2e-163) {
		tmp = t_m * (sqrt(2.0) * sqrt(((-1.0 / t_4) + (x / t_4))));
	} else if (t_m <= 3.4e+47) {
		tmp = t_2 / sqrt((fma(2.0, (pow(t_m, 2.0) / x), fma(2.0, pow(t_m, 2.0), (pow(l_m, 2.0) / x))) + (fma(2.0, pow(t_m, 2.0), pow(l_m, 2.0)) / x)));
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0))
	t_4 = fma(l_m, l_m, (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.8e-226)
		tmp = Float64(Float64(t_m / (x ^ -0.5)) / l_m);
	elseif (t_m <= 1.25e-188)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(x * sqrt(2.0))))) + t_2)));
	elseif (t_m <= 7.2e-163)
		tmp = Float64(t_m * Float64(sqrt(2.0) * sqrt(Float64(Float64(-1.0 / t_4) + Float64(x / t_4)))));
	elseif (t_m <= 3.4e+47)
		tmp = Float64(t_2 / sqrt(Float64(fma(2.0, Float64((t_m ^ 2.0) / x), fma(2.0, (t_m ^ 2.0), Float64((l_m ^ 2.0) / x))) + Float64(fma(2.0, (t_m ^ 2.0), (l_m ^ 2.0)) / x))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(l$95$m * l$95$m + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.8e-226], N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.25e-188], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 7.2e-163], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / t$95$4), $MachinePrecision] + N[(x / t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.4e+47], N[(t$95$2 / N[Sqrt[N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision] + N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 1.79999999999999997e-226

   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%

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

   5. Taylor expanded in x around inf 58.2%

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

   6. Taylor expanded in l around inf 20.8%

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

      1. *-commutative 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. pow1/2 20.8%

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

      3. inv-pow 20.8%

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

      4. pow-pow 20.8%

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

      5. metadata-eval 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

   12. Simplified 20.8%

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

       1. associate-*r* 20.8%

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

       2. associate-*r/ 20.8%

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

       3. times-frac 20.9%

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

       4. sqrt-undiv 20.9%

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

       5. metadata-eval 20.9%

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

       6. metadata-eval 20.9%

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

       7. metadata-eval 20.9%

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

       8. times-frac 20.9%

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

       9. *-un-lft-identity 20.9%

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

       10. *-un-lft-identity 20.9%

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

   14. Applied egg-rr 20.9%

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

       1. *-lft-identity 20.9%

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

       2. associate-*r/ 20.9%

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

       3. times-frac 20.9%

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

       4. associate-*l/ 20.9%

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

       5. *-lft-identity 20.9%

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

   16. Simplified 20.9%

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

   if 1.79999999999999997e-226 < t < 1.25e-188

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in x around inf 68.3%

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

   if 1.25e-188 < t < 7.1999999999999996e-163

   1. Initial program 2.4%

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

   3. Simplified 3.4%

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

      1. clear-num 5.4%

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

      2. inv-pow 5.4%

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

      3. pow2 5.4%

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

      4. pow2 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. unpow-1 5.4%

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

      2. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in t around 0 6.6%

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

       1. associate-*l* 6.6%

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

       2. associate-/l* 2.4%

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

       3. +-commutative 2.4%

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

       4. sub-neg 2.4%

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

       5. metadata-eval 2.4%

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

       6. fma-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in x around 0 99.0%

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

       1. +-commutative 99.0%

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

       2. sub-neg 99.0%

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

       3. neg-mul-1 99.0%

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

       4. distribute-neg-out 99.0%

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

       5. *-lft-identity 99.0%

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

       6. metadata-eval 99.0%

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

       7. cancel-sign-sub-inv 99.0%

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

       8. distribute-neg-frac2 99.0%

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

       9. distribute-neg-frac 99.0%

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

       10. metadata-eval 99.0%

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

       11. unpow2 99.0%

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

       12. fma-neg 99.0%

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

       13. neg-mul-1 99.0%

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

       14. remove-double-neg 99.0%

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

       15. mul-1-neg 99.0%

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

       16. distribute-neg-frac2 99.0%

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

   14. Simplified 99.0%

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

   if 7.1999999999999996e-163 < t < 3.3999999999999998e47

   1. Initial program 51.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around inf 84.1%

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

      1. expm1-log1p-u 75.2%

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

      2. expm1-undefine 69.6%

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

      3. fma-define 69.6%

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

   7. Applied egg-rr 69.6%

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

      1. expm1-define 75.2%

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

   9. Simplified 75.2%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}\right)\right)}}} \]
   10. Step-by-step derivation

       1. associate-*r/ 75.4%

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

       2. cancel-sign-sub-inv 75.4%

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

       3. fma-define 75.4%

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

       4. fma-define 75.4%

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

       5. metadata-eval 75.4%

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

       6. *-un-lft-identity 75.4%

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

       7. expm1-log1p-u 84.2%

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

   11. Applied egg-rr 84.2%

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

   if 3.3999999999999998e47 < t

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in t around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 97.7%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 97.7%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 97.7%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 97.7%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 97.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)} + \frac{x}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)}}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{{t}^{2}}{x}, \mathsf{fma}\left(2, {t}^{2}, \frac{{\ell}^{2}}{x}\right)\right) + \frac{\mathsf{fma}\left(2, {t}^{2}, {\ell}^{2}\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(l\_m, l\_m, {l\_m}^{2}\right)\\ t_3 := 2 - 4 \cdot \frac{-1}{x}\\ t_4 := \frac{{l\_m}^{2}}{x}\\ t_5 := 2 \cdot {t\_m}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \left(\frac{t\_4 + t\_4}{t\_m} \cdot \sqrt{\frac{1}{t\_3}}\right) + t\_m \cdot \sqrt{t\_3}}\\ \mathbf{elif}\;t\_m \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{t\_2} + \frac{x}{t\_2}}\right)\\ \mathbf{elif}\;t\_m \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_5 + t\_4\right)\right) + \frac{t\_5 + {l\_m}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma l_m l_m (pow l_m 2.0)))
        (t_3 (- 2.0 (* 4.0 (/ -1.0 x))))
        (t_4 (/ (pow l_m 2.0) x))
        (t_5 (* 2.0 (pow t_m 2.0))))
   (*
    t_s
    (if (<= t_m 3.8e-226)
      (/ (/ t_m (pow x -0.5)) l_m)
      (if (<= t_m 7.5e-189)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (* (/ (+ t_4 t_4) t_m) (sqrt (/ 1.0 t_3))))
           (* t_m (sqrt t_3)))))
        (if (<= t_m 3.35e-164)
          (* t_m (* (sqrt 2.0) (sqrt (+ (/ -1.0 t_2) (/ x t_2)))))
          (if (<= t_m 2.2e+47)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_5 t_4))
                (/ (+ t_5 (pow l_m 2.0)) x)))))
            (+
             1.0
             (+
              (/ 0.5 (pow x 2.0))
              (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = fma(l_m, l_m, pow(l_m, 2.0));
	double t_3 = 2.0 - (4.0 * (-1.0 / x));
	double t_4 = pow(l_m, 2.0) / x;
	double t_5 = 2.0 * pow(t_m, 2.0);
	double tmp;
	if (t_m <= 3.8e-226) {
		tmp = (t_m / pow(x, -0.5)) / l_m;
	} else if (t_m <= 7.5e-189) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((t_4 + t_4) / t_m) * sqrt((1.0 / t_3)))) + (t_m * sqrt(t_3))));
	} else if (t_m <= 3.35e-164) {
		tmp = t_m * (sqrt(2.0) * sqrt(((-1.0 / t_2) + (x / t_2))));
	} else if (t_m <= 2.2e+47) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_5 + t_4)) + ((t_5 + pow(l_m, 2.0)) / x))));
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = fma(l_m, l_m, (l_m ^ 2.0))
	t_3 = Float64(2.0 - Float64(4.0 * Float64(-1.0 / x)))
	t_4 = Float64((l_m ^ 2.0) / x)
	t_5 = Float64(2.0 * (t_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 3.8e-226)
		tmp = Float64(Float64(t_m / (x ^ -0.5)) / l_m);
	elseif (t_m <= 7.5e-189)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(t_4 + t_4) / t_m) * sqrt(Float64(1.0 / t_3)))) + Float64(t_m * sqrt(t_3)))));
	elseif (t_m <= 3.35e-164)
		tmp = Float64(t_m * Float64(sqrt(2.0) * sqrt(Float64(Float64(-1.0 / t_2) + Float64(x / t_2)))));
	elseif (t_m <= 2.2e+47)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_5 + t_4)) + Float64(Float64(t_5 + (l_m ^ 2.0)) / x)))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(l$95$m * l$95$m + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 - N[(4.0 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 3.8e-226], N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 7.5e-189], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(t$95$4 + t$95$4), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 3.35e-164], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / t$95$2), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.2e+47], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 + t$95$4), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$5 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < 3.79999999999999981e-226

   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%

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

   5. Taylor expanded in x around inf 58.2%

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

   6. Taylor expanded in l around inf 20.8%

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

      1. *-commutative 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. pow1/2 20.8%

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

      3. inv-pow 20.8%

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

      4. pow-pow 20.8%

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

      5. metadata-eval 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

   12. Simplified 20.8%

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

       1. associate-*r* 20.8%

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

       2. associate-*r/ 20.8%

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

       3. times-frac 20.9%

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

       4. sqrt-undiv 20.9%

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

       5. metadata-eval 20.9%

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

       6. metadata-eval 20.9%

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

       7. metadata-eval 20.9%

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

       8. times-frac 20.9%

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

       9. *-un-lft-identity 20.9%

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

       10. *-un-lft-identity 20.9%

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

   14. Applied egg-rr 20.9%

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

       1. *-lft-identity 20.9%

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

       2. associate-*r/ 20.9%

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

       3. times-frac 20.9%

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

       4. associate-*l/ 20.9%

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

       5. *-lft-identity 20.9%

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

   16. Simplified 20.9%

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

   if 3.79999999999999981e-226 < t < 7.50000000000000042e-189

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in x around inf 35.2%

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

   6. Taylor expanded in t around inf 68.3%

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

   if 7.50000000000000042e-189 < t < 3.35e-164

   1. Initial program 2.4%

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

   3. Simplified 3.4%

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

      1. clear-num 5.4%

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

      2. inv-pow 5.4%

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

      3. pow2 5.4%

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

      4. pow2 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. unpow-1 5.4%

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

      2. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in t around 0 6.6%

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

       1. associate-*l* 6.6%

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

       2. associate-/l* 2.4%

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

       3. +-commutative 2.4%

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

       4. sub-neg 2.4%

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

       5. metadata-eval 2.4%

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

       6. fma-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in x around 0 99.0%

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

       1. +-commutative 99.0%

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

       2. sub-neg 99.0%

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

       3. neg-mul-1 99.0%

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

       4. distribute-neg-out 99.0%

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

       5. *-lft-identity 99.0%

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

       6. metadata-eval 99.0%

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

       7. cancel-sign-sub-inv 99.0%

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

       8. distribute-neg-frac2 99.0%

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

       9. distribute-neg-frac 99.0%

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

       10. metadata-eval 99.0%

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

       11. unpow2 99.0%

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

       12. fma-neg 99.0%

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

       13. neg-mul-1 99.0%

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

       14. remove-double-neg 99.0%

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

       15. mul-1-neg 99.0%

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

       16. distribute-neg-frac2 99.0%

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

   14. Simplified 99.0%

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

   if 3.35e-164 < t < 2.1999999999999999e47

   1. Initial program 51.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around inf 84.1%

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

   if 2.1999999999999999e47 < t

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in t around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 97.7%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 97.7%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 97.7%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 97.7%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 97.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-189}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 - 4 \cdot \frac{-1}{x}}}\right) + t \cdot \sqrt{2 - 4 \cdot \frac{-1}{x}}}\\ \mathbf{elif}\;t \leq 3.35 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)} + \frac{x}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)}}\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 \cdot {t\_m}^{2}\\ t_3 := t\_2 + {l\_m}^{2}\\ t_4 := \mathsf{fma}\left(l\_m, l\_m, {l\_m}^{2}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \frac{t\_3 + t\_3}{t\_m \cdot \left(x \cdot \sqrt{2}\right)} + t\_m \cdot \sqrt{2}}\\ \mathbf{elif}\;t\_m \leq 1.08 \cdot 10^{-162}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{t\_4} + \frac{x}{t\_4}}\right)\\ \mathbf{elif}\;t\_m \leq 9 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(t\_2 + \frac{{l\_m}^{2}}{x}\right)\right) + \frac{t\_3}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* 2.0 (pow t_m 2.0)))
        (t_3 (+ t_2 (pow l_m 2.0)))
        (t_4 (fma l_m l_m (pow l_m 2.0))))
   (*
    t_s
    (if (<= t_m 4.1e-226)
      (/ (/ t_m (pow x -0.5)) l_m)
      (if (<= t_m 1.1e-188)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (/ (+ t_3 t_3) (* t_m (* x (sqrt 2.0)))))
           (* t_m (sqrt 2.0)))))
        (if (<= t_m 1.08e-162)
          (* t_m (* (sqrt 2.0) (sqrt (+ (/ -1.0 t_4) (/ x t_4)))))
          (if (<= t_m 9e+46)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (+ (* 2.0 (/ (pow t_m 2.0) x)) (+ t_2 (/ (pow l_m 2.0) x)))
                (/ t_3 x)))))
            (+
             1.0
             (+
              (/ 0.5 (pow x 2.0))
              (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 * pow(t_m, 2.0);
	double t_3 = t_2 + pow(l_m, 2.0);
	double t_4 = fma(l_m, l_m, pow(l_m, 2.0));
	double tmp;
	if (t_m <= 4.1e-226) {
		tmp = (t_m / pow(x, -0.5)) / l_m;
	} else if (t_m <= 1.1e-188) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * ((t_3 + t_3) / (t_m * (x * sqrt(2.0))))) + (t_m * sqrt(2.0))));
	} else if (t_m <= 1.08e-162) {
		tmp = t_m * (sqrt(2.0) * sqrt(((-1.0 / t_4) + (x / t_4))));
	} else if (t_m <= 9e+46) {
		tmp = sqrt(2.0) * (t_m / sqrt((((2.0 * (pow(t_m, 2.0) / x)) + (t_2 + (pow(l_m, 2.0) / x))) + (t_3 / x))));
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 * (t_m ^ 2.0))
	t_3 = Float64(t_2 + (l_m ^ 2.0))
	t_4 = fma(l_m, l_m, (l_m ^ 2.0))
	tmp = 0.0
	if (t_m <= 4.1e-226)
		tmp = Float64(Float64(t_m / (x ^ -0.5)) / l_m);
	elseif (t_m <= 1.1e-188)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(t_3 + t_3) / Float64(t_m * Float64(x * sqrt(2.0))))) + Float64(t_m * sqrt(2.0)))));
	elseif (t_m <= 1.08e-162)
		tmp = Float64(t_m * Float64(sqrt(2.0) * sqrt(Float64(Float64(-1.0 / t_4) + Float64(x / t_4)))));
	elseif (t_m <= 9e+46)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(t_2 + Float64((l_m ^ 2.0) / x))) + Float64(t_3 / x)))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(l$95$m * l$95$m + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.1e-226], N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.1e-188], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(t$95$3 + t$95$3), $MachinePrecision] / N[(t$95$m * N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.08e-162], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / t$95$4), $MachinePrecision] + N[(x / t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 9e+46], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 4.10000000000000037e-226

   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%

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

   5. Taylor expanded in x around inf 58.2%

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

   6. Taylor expanded in l around inf 20.8%

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

      1. *-commutative 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. pow1/2 20.8%

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

      3. inv-pow 20.8%

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

      4. pow-pow 20.8%

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

      5. metadata-eval 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

   12. Simplified 20.8%

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

       1. associate-*r* 20.8%

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

       2. associate-*r/ 20.8%

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

       3. times-frac 20.9%

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

       4. sqrt-undiv 20.9%

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

       5. metadata-eval 20.9%

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

       6. metadata-eval 20.9%

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

       7. metadata-eval 20.9%

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

       8. times-frac 20.9%

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

       9. *-un-lft-identity 20.9%

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

       10. *-un-lft-identity 20.9%

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

   14. Applied egg-rr 20.9%

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

       1. *-lft-identity 20.9%

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

       2. associate-*r/ 20.9%

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

       3. times-frac 20.9%

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

       4. associate-*l/ 20.9%

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

       5. *-lft-identity 20.9%

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

   16. Simplified 20.9%

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

   if 4.10000000000000037e-226 < t < 1.1e-188

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in x around inf 68.3%

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

   if 1.1e-188 < t < 1.08000000000000006e-162

   1. Initial program 2.4%

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

   3. Simplified 3.4%

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

      1. clear-num 5.4%

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

      2. inv-pow 5.4%

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

      3. pow2 5.4%

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

      4. pow2 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. unpow-1 5.4%

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

      2. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in t around 0 6.6%

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

       1. associate-*l* 6.6%

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

       2. associate-/l* 2.4%

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

       3. +-commutative 2.4%

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

       4. sub-neg 2.4%

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

       5. metadata-eval 2.4%

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

       6. fma-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in x around 0 99.0%

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

       1. +-commutative 99.0%

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

       2. sub-neg 99.0%

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

       3. neg-mul-1 99.0%

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

       4. distribute-neg-out 99.0%

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

       5. *-lft-identity 99.0%

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

       6. metadata-eval 99.0%

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

       7. cancel-sign-sub-inv 99.0%

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

       8. distribute-neg-frac2 99.0%

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

       9. distribute-neg-frac 99.0%

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

       10. metadata-eval 99.0%

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

       11. unpow2 99.0%

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

       12. fma-neg 99.0%

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

       13. neg-mul-1 99.0%

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

       14. remove-double-neg 99.0%

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

       15. mul-1-neg 99.0%

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

       16. distribute-neg-frac2 99.0%

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

   14. Simplified 99.0%

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

   if 1.08000000000000006e-162 < t < 9.00000000000000019e46

   1. Initial program 51.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around inf 84.1%

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

   if 9.00000000000000019e46 < t

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in t around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 97.7%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 97.7%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 97.7%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 97.7%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 97.7%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \frac{\left(2 \cdot {t}^{2} + {\ell}^{2}\right) + \left(2 \cdot {t}^{2} + {\ell}^{2}\right)}{t \cdot \left(x \cdot \sqrt{2}\right)} + t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-162}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)} + \frac{x}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)}}\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) + \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \mathsf{fma}\left(l\_m, l\_m, {l\_m}^{2}\right)\\ t_3 := 2 - 4 \cdot \frac{-1}{x}\\ t_4 := \frac{{l\_m}^{2}}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.6 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \left(\frac{t\_4 + t\_4}{t\_m} \cdot \sqrt{\frac{1}{t\_3}}\right) + t\_m \cdot \sqrt{t\_3}}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{-164}:\\ \;\;\;\;t\_m \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{t\_2} + \frac{x}{t\_2}}\right)\\ \mathbf{elif}\;t\_m \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{t\_4 + \left(2 \cdot \frac{{t\_m}^{2}}{x} + \left(2 \cdot {t\_m}^{2} + t\_4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (fma l_m l_m (pow l_m 2.0)))
        (t_3 (- 2.0 (* 4.0 (/ -1.0 x))))
        (t_4 (/ (pow l_m 2.0) x)))
   (*
    t_s
    (if (<= t_m 1.55e-225)
      (/ (/ t_m (pow x -0.5)) l_m)
      (if (<= t_m 1.6e-188)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (* (/ (+ t_4 t_4) t_m) (sqrt (/ 1.0 t_3))))
           (* t_m (sqrt t_3)))))
        (if (<= t_m 5.2e-164)
          (* t_m (* (sqrt 2.0) (sqrt (+ (/ -1.0 t_2) (/ x t_2)))))
          (if (<= t_m 4.2e+47)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                t_4
                (+
                 (* 2.0 (/ (pow t_m 2.0) x))
                 (+ (* 2.0 (pow t_m 2.0)) t_4))))))
            (+
             1.0
             (+
              (/ 0.5 (pow x 2.0))
              (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = fma(l_m, l_m, pow(l_m, 2.0));
	double t_3 = 2.0 - (4.0 * (-1.0 / x));
	double t_4 = pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 1.55e-225) {
		tmp = (t_m / pow(x, -0.5)) / l_m;
	} else if (t_m <= 1.6e-188) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((t_4 + t_4) / t_m) * sqrt((1.0 / t_3)))) + (t_m * sqrt(t_3))));
	} else if (t_m <= 5.2e-164) {
		tmp = t_m * (sqrt(2.0) * sqrt(((-1.0 / t_2) + (x / t_2))));
	} else if (t_m <= 4.2e+47) {
		tmp = sqrt(2.0) * (t_m / sqrt((t_4 + ((2.0 * (pow(t_m, 2.0) / x)) + ((2.0 * pow(t_m, 2.0)) + t_4)))));
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = fma(l_m, l_m, (l_m ^ 2.0))
	t_3 = Float64(2.0 - Float64(4.0 * Float64(-1.0 / x)))
	t_4 = Float64((l_m ^ 2.0) / x)
	tmp = 0.0
	if (t_m <= 1.55e-225)
		tmp = Float64(Float64(t_m / (x ^ -0.5)) / l_m);
	elseif (t_m <= 1.6e-188)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(t_4 + t_4) / t_m) * sqrt(Float64(1.0 / t_3)))) + Float64(t_m * sqrt(t_3)))));
	elseif (t_m <= 5.2e-164)
		tmp = Float64(t_m * Float64(sqrt(2.0) * sqrt(Float64(Float64(-1.0 / t_2) + Float64(x / t_2)))));
	elseif (t_m <= 4.2e+47)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(t_4 + Float64(Float64(2.0 * Float64((t_m ^ 2.0) / x)) + Float64(Float64(2.0 * (t_m ^ 2.0)) + t_4))))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(l$95$m * l$95$m + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 - N[(4.0 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-225], N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.6e-188], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(t$95$4 + t$95$4), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e-164], N[(t$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(-1.0 / t$95$2), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.2e+47], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(t$95$4 + N[(N[(2.0 * N[(N[Power[t$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 1.54999999999999998e-225

   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%

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

   5. Taylor expanded in x around inf 58.2%

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

   6. Taylor expanded in l around inf 20.8%

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

      1. *-commutative 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. pow1/2 20.8%

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

      3. inv-pow 20.8%

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

      4. pow-pow 20.8%

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

      5. metadata-eval 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

   12. Simplified 20.8%

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

       1. associate-*r* 20.8%

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

       2. associate-*r/ 20.8%

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

       3. times-frac 20.9%

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

       4. sqrt-undiv 20.9%

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

       5. metadata-eval 20.9%

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

       6. metadata-eval 20.9%

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

       7. metadata-eval 20.9%

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

       8. times-frac 20.9%

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

       9. *-un-lft-identity 20.9%

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

       10. *-un-lft-identity 20.9%

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

   14. Applied egg-rr 20.9%

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

       1. *-lft-identity 20.9%

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

       2. associate-*r/ 20.9%

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

       3. times-frac 20.9%

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

       4. associate-*l/ 20.9%

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

       5. *-lft-identity 20.9%

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

   16. Simplified 20.9%

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

   if 1.54999999999999998e-225 < t < 1.60000000000000011e-188

   1. Initial program 2.8%

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

   3. Simplified 2.8%

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

   5. Taylor expanded in x around inf 35.2%

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

   6. Taylor expanded in t around inf 68.3%

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

   if 1.60000000000000011e-188 < t < 5.2000000000000003e-164

   1. Initial program 2.4%

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

   3. Simplified 3.4%

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

      1. clear-num 5.4%

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

      2. inv-pow 5.4%

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

      3. pow2 5.4%

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

      4. pow2 5.4%

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

   6. Applied egg-rr 5.4%

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

      1. unpow-1 5.4%

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

      2. +-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in t around 0 6.6%

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

       1. associate-*l* 6.6%

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

       2. associate-/l* 2.4%

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

       3. +-commutative 2.4%

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

       4. sub-neg 2.4%

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

       5. metadata-eval 2.4%

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

       6. fma-neg 2.4%

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

   11. Simplified 2.4%

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

   12. Taylor expanded in x around 0 99.0%

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

       1. +-commutative 99.0%

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

       2. sub-neg 99.0%

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

       3. neg-mul-1 99.0%

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

       4. distribute-neg-out 99.0%

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

       5. *-lft-identity 99.0%

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

       6. metadata-eval 99.0%

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

       7. cancel-sign-sub-inv 99.0%

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

       8. distribute-neg-frac2 99.0%

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

       9. distribute-neg-frac 99.0%

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

       10. metadata-eval 99.0%

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

       11. unpow2 99.0%

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

       12. fma-neg 99.0%

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

       13. neg-mul-1 99.0%

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

       14. remove-double-neg 99.0%

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

       15. mul-1-neg 99.0%

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

       16. distribute-neg-frac2 99.0%

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

   14. Simplified 99.0%

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

   if 5.2000000000000003e-164 < t < 4.2e47

   1. Initial program 51.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around inf 84.1%

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

   6. Taylor expanded in t around 0 83.1%

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

   if 4.2e47 < t

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in t around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 97.7%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 97.7%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 97.7%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 97.7%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 97.7%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 - 4 \cdot \frac{-1}{x}}}\right) + t \cdot \sqrt{2 - 4 \cdot \frac{-1}{x}}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-164}:\\ \;\;\;\;t \cdot \left(\sqrt{2} \cdot \sqrt{\frac{-1}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)} + \frac{x}{\mathsf{fma}\left(\ell, \ell, {\ell}^{2}\right)}}\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{{\ell}^{2}}{x} + \left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := 2 - 4 \cdot \frac{-1}{x}\\ t_3 := \frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ t_4 := \frac{{l\_m}^{2}}{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 1.35 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{0.5 \cdot \left(\frac{t\_4 + t\_4}{t\_m} \cdot \sqrt{\frac{1}{t\_2}}\right) + t\_m \cdot \sqrt{t\_2}}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;{\left(\sqrt{t\_3}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 7.5 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{2 \cdot {t\_m}^{2} + {l\_m}^{2}}{x} + {t\_m}^{2} \cdot \left(2 + \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (- 2.0 (* 4.0 (/ -1.0 x))))
        (t_3 (/ (/ t_m (pow x -0.5)) l_m))
        (t_4 (/ (pow l_m 2.0) x)))
   (*
    t_s
    (if (<= t_m 1.2e-226)
      t_3
      (if (<= t_m 1.35e-122)
        (*
         (sqrt 2.0)
         (/
          t_m
          (+
           (* 0.5 (* (/ (+ t_4 t_4) t_m) (sqrt (/ 1.0 t_2))))
           (* t_m (sqrt t_2)))))
        (if (<= t_m 2.4e-98)
          (pow (sqrt t_3) 2.0)
          (if (<= t_m 7.5e+46)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (/ (+ (* 2.0 (pow t_m 2.0)) (pow l_m 2.0)) x)
                (* (pow t_m 2.0) (+ 2.0 (/ 2.0 x)))))))
            (+
             1.0
             (+
              (/ 0.5 (pow x 2.0))
              (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x)))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 - (4.0 * (-1.0 / x));
	double t_3 = (t_m / pow(x, -0.5)) / l_m;
	double t_4 = pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 1.2e-226) {
		tmp = t_3;
	} else if (t_m <= 1.35e-122) {
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((t_4 + t_4) / t_m) * sqrt((1.0 / t_2)))) + (t_m * sqrt(t_2))));
	} else if (t_m <= 2.4e-98) {
		tmp = pow(sqrt(t_3), 2.0);
	} else if (t_m <= 7.5e+46) {
		tmp = sqrt(2.0) * (t_m / sqrt(((((2.0 * pow(t_m, 2.0)) + pow(l_m, 2.0)) / x) + (pow(t_m, 2.0) * (2.0 + (2.0 / x))))));
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_2 = 2.0d0 - (4.0d0 * ((-1.0d0) / x))
    t_3 = (t_m / (x ** (-0.5d0))) / l_m
    t_4 = (l_m ** 2.0d0) / x
    if (t_m <= 1.2d-226) then
        tmp = t_3
    else if (t_m <= 1.35d-122) then
        tmp = sqrt(2.0d0) * (t_m / ((0.5d0 * (((t_4 + t_4) / t_m) * sqrt((1.0d0 / t_2)))) + (t_m * sqrt(t_2))))
    else if (t_m <= 2.4d-98) then
        tmp = sqrt(t_3) ** 2.0d0
    else if (t_m <= 7.5d+46) then
        tmp = sqrt(2.0d0) * (t_m / sqrt(((((2.0d0 * (t_m ** 2.0d0)) + (l_m ** 2.0d0)) / x) + ((t_m ** 2.0d0) * (2.0d0 + (2.0d0 / x))))))
    else
        tmp = 1.0d0 + ((0.5d0 / (x ** 2.0d0)) + (((-0.5d0) / (x ** 3.0d0)) + ((-1.0d0) / x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = 2.0 - (4.0 * (-1.0 / x));
	double t_3 = (t_m / Math.pow(x, -0.5)) / l_m;
	double t_4 = Math.pow(l_m, 2.0) / x;
	double tmp;
	if (t_m <= 1.2e-226) {
		tmp = t_3;
	} else if (t_m <= 1.35e-122) {
		tmp = Math.sqrt(2.0) * (t_m / ((0.5 * (((t_4 + t_4) / t_m) * Math.sqrt((1.0 / t_2)))) + (t_m * Math.sqrt(t_2))));
	} else if (t_m <= 2.4e-98) {
		tmp = Math.pow(Math.sqrt(t_3), 2.0);
	} else if (t_m <= 7.5e+46) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((((2.0 * Math.pow(t_m, 2.0)) + Math.pow(l_m, 2.0)) / x) + (Math.pow(t_m, 2.0) * (2.0 + (2.0 / x))))));
	} else {
		tmp = 1.0 + ((0.5 / Math.pow(x, 2.0)) + ((-0.5 / Math.pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = 2.0 - (4.0 * (-1.0 / x))
	t_3 = (t_m / math.pow(x, -0.5)) / l_m
	t_4 = math.pow(l_m, 2.0) / x
	tmp = 0
	if t_m <= 1.2e-226:
		tmp = t_3
	elif t_m <= 1.35e-122:
		tmp = math.sqrt(2.0) * (t_m / ((0.5 * (((t_4 + t_4) / t_m) * math.sqrt((1.0 / t_2)))) + (t_m * math.sqrt(t_2))))
	elif t_m <= 2.4e-98:
		tmp = math.pow(math.sqrt(t_3), 2.0)
	elif t_m <= 7.5e+46:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((((2.0 * math.pow(t_m, 2.0)) + math.pow(l_m, 2.0)) / x) + (math.pow(t_m, 2.0) * (2.0 + (2.0 / x))))))
	else:
		tmp = 1.0 + ((0.5 / math.pow(x, 2.0)) + ((-0.5 / math.pow(x, 3.0)) + (-1.0 / x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(2.0 - Float64(4.0 * Float64(-1.0 / x)))
	t_3 = Float64(Float64(t_m / (x ^ -0.5)) / l_m)
	t_4 = Float64((l_m ^ 2.0) / x)
	tmp = 0.0
	if (t_m <= 1.2e-226)
		tmp = t_3;
	elseif (t_m <= 1.35e-122)
		tmp = Float64(sqrt(2.0) * Float64(t_m / Float64(Float64(0.5 * Float64(Float64(Float64(t_4 + t_4) / t_m) * sqrt(Float64(1.0 / t_2)))) + Float64(t_m * sqrt(t_2)))));
	elseif (t_m <= 2.4e-98)
		tmp = sqrt(t_3) ^ 2.0;
	elseif (t_m <= 7.5e+46)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0)) / x) + Float64((t_m ^ 2.0) * Float64(2.0 + Float64(2.0 / x)))))));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = 2.0 - (4.0 * (-1.0 / x));
	t_3 = (t_m / (x ^ -0.5)) / l_m;
	t_4 = (l_m ^ 2.0) / x;
	tmp = 0.0;
	if (t_m <= 1.2e-226)
		tmp = t_3;
	elseif (t_m <= 1.35e-122)
		tmp = sqrt(2.0) * (t_m / ((0.5 * (((t_4 + t_4) / t_m) * sqrt((1.0 / t_2)))) + (t_m * sqrt(t_2))));
	elseif (t_m <= 2.4e-98)
		tmp = sqrt(t_3) ^ 2.0;
	elseif (t_m <= 7.5e+46)
		tmp = sqrt(2.0) * (t_m / sqrt(((((2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0)) / x) + ((t_m ^ 2.0) * (2.0 + (2.0 / x))))));
	else
		tmp = 1.0 + ((0.5 / (x ^ 2.0)) + ((-0.5 / (x ^ 3.0)) + (-1.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(2.0 - N[(4.0 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[l$95$m, 2.0], $MachinePrecision] / x), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-226], t$95$3, If[LessEqual[t$95$m, 1.35e-122], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[(N[(0.5 * N[(N[(N[(t$95$4 + t$95$4), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e-98], N[Power[N[Sqrt[t$95$3], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[t$95$m, 7.5e+46], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if t < 1.2e-226

   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%

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

   5. Taylor expanded in x around inf 58.2%

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

   6. Taylor expanded in l around inf 20.8%

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

      1. *-commutative 20.8%

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

   8. Simplified 20.8%

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

      1. *-un-lft-identity 20.8%

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

      2. pow1/2 20.8%

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

      3. inv-pow 20.8%

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

      4. pow-pow 20.8%

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

      5. metadata-eval 20.8%

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

   10. Applied egg-rr 20.8%

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

       1. *-lft-identity 20.8%

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

   12. Simplified 20.8%

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

       1. associate-*r* 20.8%

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

       2. associate-*r/ 20.8%

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

       3. times-frac 20.9%

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

       4. sqrt-undiv 20.9%

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

       5. metadata-eval 20.9%

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

       6. metadata-eval 20.9%

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

       7. metadata-eval 20.9%

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

       8. times-frac 20.9%

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

       9. *-un-lft-identity 20.9%

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

       10. *-un-lft-identity 20.9%

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

   14. Applied egg-rr 20.9%

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

       1. *-lft-identity 20.9%

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

       2. associate-*r/ 20.9%

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

       3. times-frac 20.9%

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

       4. associate-*l/ 20.9%

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

       5. *-lft-identity 20.9%

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

   16. Simplified 20.9%

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

   if 1.2e-226 < t < 1.35000000000000005e-122

   1. Initial program 27.9%

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

   3. Simplified 27.9%

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

   5. Taylor expanded in x around inf 75.3%

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

   6. Taylor expanded in t around inf 61.1%

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

   if 1.35000000000000005e-122 < t < 2.40000000000000005e-98

   1. Initial program 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in x around inf 80.0%

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

   6. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

   8. Simplified 3.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*l* 0.0%

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

      5. pow1/2 0.0%

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

      6. inv-pow 0.0%

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

      7. pow-pow 0.0%

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

      8. metadata-eval 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. pow1 0.0%

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

       2. pow1 0.0%

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

       3. times-frac 0.0%

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

       4. sqrt-undiv 0.0%

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

       5. metadata-eval 0.0%

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

       6. metadata-eval 0.0%

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

       7. sqrt-prod 0.0%

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

       8. metadata-eval 0.0%

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

   12. Applied egg-rr 0.0%

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

       1. *-lft-identity 0.0%

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

       2. *-lft-identity 0.0%

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

       3. associate-*r/ 0.0%

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

       4. times-frac 0.0%

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

       5. associate-*l/ 0.0%

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

       6. *-lft-identity 0.0%

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

   14. Simplified 0.0%

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

   if 2.40000000000000005e-98 < t < 7.5000000000000003e46

   1. Initial program 55.4%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in x around inf 82.7%

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

   6. Taylor expanded in t around inf 68.8%

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

      1. associate-*r/ 68.8%

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

      2. metadata-eval 68.8%

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

   8. Simplified 68.8%

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

   if 7.5000000000000003e46 < t

   1. Initial program 30.6%

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

   3. Simplified 30.6%

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

   5. Taylor expanded in t around inf 97.6%

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

      1. +-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. metadata-eval 97.6%

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

      4. +-commutative 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in x around inf 97.7%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 97.7%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 97.7%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 97.7%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 97.7%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 97.7%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 97.7%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 97.7%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 97.7%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 97.7%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-122}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{0.5 \cdot \left(\frac{\frac{{\ell}^{2}}{x} + \frac{{\ell}^{2}}{x}}{t} \cdot \sqrt{\frac{1}{2 - 4 \cdot \frac{-1}{x}}}\right) + t \cdot \sqrt{2 - 4 \cdot \frac{-1}{x}}}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;{\left(\sqrt{\frac{\frac{t}{{x}^{-0.5}}}{\ell}}\right)}^{2}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+46}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.65 \cdot 10^{-199}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{-146}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t\_m}{\sqrt{\frac{2 \cdot {t\_m}^{2} + {l\_m}^{2}}{x} + {t\_m}^{2} \cdot \left(2 + \frac{2}{x}\right)}}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{-98}:\\ \;\;\;\;{\left(\sqrt{t\_2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (/ t_m (pow x -0.5)) l_m)))
   (*
    t_s
    (if (<= t_m 1.4e-225)
      t_2
      (if (<= t_m 1.65e-199)
        1.0
        (if (<= t_m 2e-146)
          t_2
          (if (<= t_m 1.1e-121)
            (*
             (sqrt 2.0)
             (/
              t_m
              (sqrt
               (+
                (/ (+ (* 2.0 (pow t_m 2.0)) (pow l_m 2.0)) x)
                (* (pow t_m 2.0) (+ 2.0 (/ 2.0 x)))))))
            (if (<= t_m 4.1e-98)
              (pow (sqrt t_2) 2.0)
              (+
               1.0
               (+
                (/ 0.5 (pow x 2.0))
                (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x))))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 1.4e-225) {
		tmp = t_2;
	} else if (t_m <= 1.65e-199) {
		tmp = 1.0;
	} else if (t_m <= 2e-146) {
		tmp = t_2;
	} else if (t_m <= 1.1e-121) {
		tmp = sqrt(2.0) * (t_m / sqrt(((((2.0 * pow(t_m, 2.0)) + pow(l_m, 2.0)) / x) + (pow(t_m, 2.0) * (2.0 + (2.0 / x))))));
	} else if (t_m <= 4.1e-98) {
		tmp = pow(sqrt(t_2), 2.0);
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / (x ** (-0.5d0))) / l_m
    if (t_m <= 1.4d-225) then
        tmp = t_2
    else if (t_m <= 1.65d-199) then
        tmp = 1.0d0
    else if (t_m <= 2d-146) then
        tmp = t_2
    else if (t_m <= 1.1d-121) then
        tmp = sqrt(2.0d0) * (t_m / sqrt(((((2.0d0 * (t_m ** 2.0d0)) + (l_m ** 2.0d0)) / x) + ((t_m ** 2.0d0) * (2.0d0 + (2.0d0 / x))))))
    else if (t_m <= 4.1d-98) then
        tmp = sqrt(t_2) ** 2.0d0
    else
        tmp = 1.0d0 + ((0.5d0 / (x ** 2.0d0)) + (((-0.5d0) / (x ** 3.0d0)) + ((-1.0d0) / x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / Math.pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 1.4e-225) {
		tmp = t_2;
	} else if (t_m <= 1.65e-199) {
		tmp = 1.0;
	} else if (t_m <= 2e-146) {
		tmp = t_2;
	} else if (t_m <= 1.1e-121) {
		tmp = Math.sqrt(2.0) * (t_m / Math.sqrt(((((2.0 * Math.pow(t_m, 2.0)) + Math.pow(l_m, 2.0)) / x) + (Math.pow(t_m, 2.0) * (2.0 + (2.0 / x))))));
	} else if (t_m <= 4.1e-98) {
		tmp = Math.pow(Math.sqrt(t_2), 2.0);
	} else {
		tmp = 1.0 + ((0.5 / Math.pow(x, 2.0)) + ((-0.5 / Math.pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / math.pow(x, -0.5)) / l_m
	tmp = 0
	if t_m <= 1.4e-225:
		tmp = t_2
	elif t_m <= 1.65e-199:
		tmp = 1.0
	elif t_m <= 2e-146:
		tmp = t_2
	elif t_m <= 1.1e-121:
		tmp = math.sqrt(2.0) * (t_m / math.sqrt(((((2.0 * math.pow(t_m, 2.0)) + math.pow(l_m, 2.0)) / x) + (math.pow(t_m, 2.0) * (2.0 + (2.0 / x))))))
	elif t_m <= 4.1e-98:
		tmp = math.pow(math.sqrt(t_2), 2.0)
	else:
		tmp = 1.0 + ((0.5 / math.pow(x, 2.0)) + ((-0.5 / math.pow(x, 3.0)) + (-1.0 / x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / (x ^ -0.5)) / l_m)
	tmp = 0.0
	if (t_m <= 1.4e-225)
		tmp = t_2;
	elseif (t_m <= 1.65e-199)
		tmp = 1.0;
	elseif (t_m <= 2e-146)
		tmp = t_2;
	elseif (t_m <= 1.1e-121)
		tmp = Float64(sqrt(2.0) * Float64(t_m / sqrt(Float64(Float64(Float64(Float64(2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0)) / x) + Float64((t_m ^ 2.0) * Float64(2.0 + Float64(2.0 / x)))))));
	elseif (t_m <= 4.1e-98)
		tmp = sqrt(t_2) ^ 2.0;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / (x ^ -0.5)) / l_m;
	tmp = 0.0;
	if (t_m <= 1.4e-225)
		tmp = t_2;
	elseif (t_m <= 1.65e-199)
		tmp = 1.0;
	elseif (t_m <= 2e-146)
		tmp = t_2;
	elseif (t_m <= 1.1e-121)
		tmp = sqrt(2.0) * (t_m / sqrt(((((2.0 * (t_m ^ 2.0)) + (l_m ^ 2.0)) / x) + ((t_m ^ 2.0) * (2.0 + (2.0 / x))))));
	elseif (t_m <= 4.1e-98)
		tmp = sqrt(t_2) ^ 2.0;
	else
		tmp = 1.0 + ((0.5 / (x ^ 2.0)) + ((-0.5 / (x ^ 3.0)) + (-1.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.4e-225], t$95$2, If[LessEqual[t$95$m, 1.65e-199], 1.0, If[LessEqual[t$95$m, 2e-146], t$95$2, If[LessEqual[t$95$m, 1.1e-121], N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / N[Sqrt[N[(N[(N[(N[(2.0 * N[Power[t$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(N[Power[t$95$m, 2.0], $MachinePrecision] * N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e-98], N[Power[N[Sqrt[t$95$2], $MachinePrecision], 2.0], $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 1.4e-225 or 1.6500000000000001e-199 < t < 2.00000000000000005e-146

   1. Initial program 30.1%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in x around inf 60.3%

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

   6. Taylor expanded in l around inf 22.0%

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

      1. *-commutative 22.0%

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

   8. Simplified 22.0%

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

      1. *-un-lft-identity 22.0%

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

      2. pow1/2 22.0%

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

      3. inv-pow 22.0%

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

      4. pow-pow 22.1%

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

      5. metadata-eval 22.1%

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

   10. Applied egg-rr 22.1%

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

       1. *-lft-identity 22.1%

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

   12. Simplified 22.1%

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

       1. associate-*r* 22.1%

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

       2. associate-*r/ 22.1%

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

       3. times-frac 22.1%

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

       4. sqrt-undiv 22.1%

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

       5. metadata-eval 22.1%

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

       6. metadata-eval 22.1%

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

       7. metadata-eval 22.1%

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

       8. times-frac 22.1%

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

       9. *-un-lft-identity 22.1%

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

       10. *-un-lft-identity 22.1%

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

   14. Applied egg-rr 22.1%

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

       1. *-lft-identity 22.1%

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

       2. associate-*r/ 22.1%

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

       3. times-frac 22.1%

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

       4. associate-*l/ 22.1%

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

       5. *-lft-identity 22.1%

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

   16. Simplified 22.1%

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

   if 1.4e-225 < t < 1.6500000000000001e-199

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2.00000000000000005e-146 < t < 1.10000000000000011e-121

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in x around inf 87.4%

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

   6. Taylor expanded in t around inf 72.4%

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

      1. associate-*r/ 72.4%

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

      2. metadata-eval 72.4%

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

   8. Simplified 72.4%

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

   if 1.10000000000000011e-121 < t < 4.0999999999999998e-98

   1. Initial program 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in x around inf 80.0%

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

   6. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

   8. Simplified 3.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*l* 0.0%

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

      5. pow1/2 0.0%

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

      6. inv-pow 0.0%

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

      7. pow-pow 0.0%

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

      8. metadata-eval 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. pow1 0.0%

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

       2. pow1 0.0%

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

       3. times-frac 0.0%

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

       4. sqrt-undiv 0.0%

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

       5. metadata-eval 0.0%

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

       6. metadata-eval 0.0%

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

       7. sqrt-prod 0.0%

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

       8. metadata-eval 0.0%

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

   12. Applied egg-rr 0.0%

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

       1. *-lft-identity 0.0%

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

       2. *-lft-identity 0.0%

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

       3. associate-*r/ 0.0%

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

       4. times-frac 0.0%

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

       5. associate-*l/ 0.0%

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

       6. *-lft-identity 0.0%

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

   14. Simplified 0.0%

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

   if 4.0999999999999998e-98 < t

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around inf 87.4%

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

      1. +-commutative 87.4%

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

      2. sub-neg 87.4%

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

      3. metadata-eval 87.4%

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

      4. +-commutative 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in x around inf 87.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 87.6%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 87.6%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 87.6%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 87.6%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 87.6%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 87.6%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 87.6%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 87.6%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 87.6%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-199}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-121}:\\ \;\;\;\;\sqrt{2} \cdot \frac{t}{\sqrt{\frac{2 \cdot {t}^{2} + {\ell}^{2}}{x} + {t}^{2} \cdot \left(2 + \frac{2}{x}\right)}}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-98}:\\ \;\;\;\;{\left(\sqrt{\frac{\frac{t}{{x}^{-0.5}}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.9 \cdot 10^{-226}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6.4 \cdot 10^{-196}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{-171}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 3.8 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;{\left(\sqrt{t\_2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (/ t_m (pow x -0.5)) l_m)))
   (*
    t_s
    (if (<= t_m 4.9e-226)
      t_2
      (if (<= t_m 6.4e-196)
        1.0
        (if (<= t_m 2.1e-171)
          t_2
          (if (<= t_m 3.8e-122)
            (+ 1.0 (/ -1.0 x))
            (if (<= t_m 2.4e-98)
              (pow (sqrt t_2) 2.0)
              (+
               1.0
               (+
                (/ 0.5 (pow x 2.0))
                (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x))))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 4.9e-226) {
		tmp = t_2;
	} else if (t_m <= 6.4e-196) {
		tmp = 1.0;
	} else if (t_m <= 2.1e-171) {
		tmp = t_2;
	} else if (t_m <= 3.8e-122) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 2.4e-98) {
		tmp = pow(sqrt(t_2), 2.0);
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / (x ** (-0.5d0))) / l_m
    if (t_m <= 4.9d-226) then
        tmp = t_2
    else if (t_m <= 6.4d-196) then
        tmp = 1.0d0
    else if (t_m <= 2.1d-171) then
        tmp = t_2
    else if (t_m <= 3.8d-122) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 2.4d-98) then
        tmp = sqrt(t_2) ** 2.0d0
    else
        tmp = 1.0d0 + ((0.5d0 / (x ** 2.0d0)) + (((-0.5d0) / (x ** 3.0d0)) + ((-1.0d0) / x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / Math.pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 4.9e-226) {
		tmp = t_2;
	} else if (t_m <= 6.4e-196) {
		tmp = 1.0;
	} else if (t_m <= 2.1e-171) {
		tmp = t_2;
	} else if (t_m <= 3.8e-122) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 2.4e-98) {
		tmp = Math.pow(Math.sqrt(t_2), 2.0);
	} else {
		tmp = 1.0 + ((0.5 / Math.pow(x, 2.0)) + ((-0.5 / Math.pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / math.pow(x, -0.5)) / l_m
	tmp = 0
	if t_m <= 4.9e-226:
		tmp = t_2
	elif t_m <= 6.4e-196:
		tmp = 1.0
	elif t_m <= 2.1e-171:
		tmp = t_2
	elif t_m <= 3.8e-122:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 2.4e-98:
		tmp = math.pow(math.sqrt(t_2), 2.0)
	else:
		tmp = 1.0 + ((0.5 / math.pow(x, 2.0)) + ((-0.5 / math.pow(x, 3.0)) + (-1.0 / x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / (x ^ -0.5)) / l_m)
	tmp = 0.0
	if (t_m <= 4.9e-226)
		tmp = t_2;
	elseif (t_m <= 6.4e-196)
		tmp = 1.0;
	elseif (t_m <= 2.1e-171)
		tmp = t_2;
	elseif (t_m <= 3.8e-122)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 2.4e-98)
		tmp = sqrt(t_2) ^ 2.0;
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / (x ^ -0.5)) / l_m;
	tmp = 0.0;
	if (t_m <= 4.9e-226)
		tmp = t_2;
	elseif (t_m <= 6.4e-196)
		tmp = 1.0;
	elseif (t_m <= 2.1e-171)
		tmp = t_2;
	elseif (t_m <= 3.8e-122)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 2.4e-98)
		tmp = sqrt(t_2) ^ 2.0;
	else
		tmp = 1.0 + ((0.5 / (x ^ 2.0)) + ((-0.5 / (x ^ 3.0)) + (-1.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4.9e-226], t$95$2, If[LessEqual[t$95$m, 6.4e-196], 1.0, If[LessEqual[t$95$m, 2.1e-171], t$95$2, If[LessEqual[t$95$m, 3.8e-122], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e-98], N[Power[N[Sqrt[t$95$2], $MachinePrecision], 2.0], $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 4.89999999999999986e-226 or 6.3999999999999999e-196 < t < 2.1e-171

   1. Initial program 30.1%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in x around inf 59.3%

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

   6. Taylor expanded in l around inf 22.1%

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

      1. *-commutative 22.1%

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

   8. Simplified 22.1%

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

      1. *-un-lft-identity 22.1%

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

      2. pow1/2 22.1%

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

      3. inv-pow 22.1%

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

      4. pow-pow 22.1%

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

      5. metadata-eval 22.1%

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

   10. Applied egg-rr 22.1%

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

       1. *-lft-identity 22.1%

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

   12. Simplified 22.1%

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

       1. associate-*r* 22.1%

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

       2. associate-*r/ 22.1%

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

       3. times-frac 22.1%

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

       4. sqrt-undiv 22.1%

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

       5. metadata-eval 22.1%

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

       6. metadata-eval 22.1%

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

       7. metadata-eval 22.1%

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

       8. times-frac 22.1%

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

       9. *-un-lft-identity 22.1%

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

       10. *-un-lft-identity 22.1%

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

   14. Applied egg-rr 22.1%

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

       1. *-lft-identity 22.1%

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

       2. associate-*r/ 22.1%

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

       3. times-frac 22.1%

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

       4. associate-*l/ 22.1%

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

       5. *-lft-identity 22.1%

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

   16. Simplified 22.1%

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

   if 4.89999999999999986e-226 < t < 6.3999999999999999e-196

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2.1e-171 < t < 3.8000000000000001e-122

   1. Initial program 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around inf 59.7%

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

      1. +-commutative 59.7%

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

      2. sub-neg 59.7%

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

      3. metadata-eval 59.7%

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

      4. +-commutative 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in x around inf 59.9%

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

   if 3.8000000000000001e-122 < t < 2.40000000000000005e-98

   1. Initial program 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in x around inf 80.0%

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

   6. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

   8. Simplified 3.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. pow2 0.0%

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

      3. associate-*r/ 0.0%

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

      4. associate-*l* 0.0%

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

      5. pow1/2 0.0%

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

      6. inv-pow 0.0%

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

      7. pow-pow 0.0%

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

      8. metadata-eval 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. pow1 0.0%

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

       2. pow1 0.0%

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

       3. times-frac 0.0%

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

       4. sqrt-undiv 0.0%

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

       5. metadata-eval 0.0%

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

       6. metadata-eval 0.0%

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

       7. sqrt-prod 0.0%

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

       8. metadata-eval 0.0%

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

   12. Applied egg-rr 0.0%

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

       1. *-lft-identity 0.0%

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

       2. *-lft-identity 0.0%

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

       3. associate-*r/ 0.0%

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

       4. times-frac 0.0%

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

       5. associate-*l/ 0.0%

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

       6. *-lft-identity 0.0%

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

   14. Simplified 0.0%

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

   if 2.40000000000000005e-98 < t

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around inf 87.4%

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

      1. +-commutative 87.4%

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

      2. sub-neg 87.4%

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

      3. metadata-eval 87.4%

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

      4. +-commutative 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in x around inf 87.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 87.6%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 87.6%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 87.6%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 87.6%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 87.6%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 87.6%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 87.6%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 87.6%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 87.6%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.9 \cdot 10^{-226}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-196}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-122}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;{\left(\sqrt{\frac{\frac{t}{{x}^{-0.5}}}{\ell}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-225}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 5.4 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 2 \cdot 10^{-171}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{t\_m}{{x}^{-0.5} \cdot l\_m}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (/ t_m (pow x -0.5)) l_m)))
   (*
    t_s
    (if (<= t_m 2.05e-225)
      t_2
      (if (<= t_m 5.4e-205)
        1.0
        (if (<= t_m 2e-171)
          t_2
          (if (<= t_m 1.5e-124)
            (+ 1.0 (/ -1.0 x))
            (if (<= t_m 4.1e-97)
              (/ t_m (* (pow x -0.5) l_m))
              (+
               1.0
               (+
                (/ 0.5 (pow x 2.0))
                (+ (/ -0.5 (pow x 3.0)) (/ -1.0 x))))))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 2.05e-225) {
		tmp = t_2;
	} else if (t_m <= 5.4e-205) {
		tmp = 1.0;
	} else if (t_m <= 2e-171) {
		tmp = t_2;
	} else if (t_m <= 1.5e-124) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 4.1e-97) {
		tmp = t_m / (pow(x, -0.5) * l_m);
	} else {
		tmp = 1.0 + ((0.5 / pow(x, 2.0)) + ((-0.5 / pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / (x ** (-0.5d0))) / l_m
    if (t_m <= 2.05d-225) then
        tmp = t_2
    else if (t_m <= 5.4d-205) then
        tmp = 1.0d0
    else if (t_m <= 2d-171) then
        tmp = t_2
    else if (t_m <= 1.5d-124) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else if (t_m <= 4.1d-97) then
        tmp = t_m / ((x ** (-0.5d0)) * l_m)
    else
        tmp = 1.0d0 + ((0.5d0 / (x ** 2.0d0)) + (((-0.5d0) / (x ** 3.0d0)) + ((-1.0d0) / x)))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / Math.pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 2.05e-225) {
		tmp = t_2;
	} else if (t_m <= 5.4e-205) {
		tmp = 1.0;
	} else if (t_m <= 2e-171) {
		tmp = t_2;
	} else if (t_m <= 1.5e-124) {
		tmp = 1.0 + (-1.0 / x);
	} else if (t_m <= 4.1e-97) {
		tmp = t_m / (Math.pow(x, -0.5) * l_m);
	} else {
		tmp = 1.0 + ((0.5 / Math.pow(x, 2.0)) + ((-0.5 / Math.pow(x, 3.0)) + (-1.0 / x)));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / math.pow(x, -0.5)) / l_m
	tmp = 0
	if t_m <= 2.05e-225:
		tmp = t_2
	elif t_m <= 5.4e-205:
		tmp = 1.0
	elif t_m <= 2e-171:
		tmp = t_2
	elif t_m <= 1.5e-124:
		tmp = 1.0 + (-1.0 / x)
	elif t_m <= 4.1e-97:
		tmp = t_m / (math.pow(x, -0.5) * l_m)
	else:
		tmp = 1.0 + ((0.5 / math.pow(x, 2.0)) + ((-0.5 / math.pow(x, 3.0)) + (-1.0 / x)))
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / (x ^ -0.5)) / l_m)
	tmp = 0.0
	if (t_m <= 2.05e-225)
		tmp = t_2;
	elseif (t_m <= 5.4e-205)
		tmp = 1.0;
	elseif (t_m <= 2e-171)
		tmp = t_2;
	elseif (t_m <= 1.5e-124)
		tmp = Float64(1.0 + Float64(-1.0 / x));
	elseif (t_m <= 4.1e-97)
		tmp = Float64(t_m / Float64((x ^ -0.5) * l_m));
	else
		tmp = Float64(1.0 + Float64(Float64(0.5 / (x ^ 2.0)) + Float64(Float64(-0.5 / (x ^ 3.0)) + Float64(-1.0 / x))));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / (x ^ -0.5)) / l_m;
	tmp = 0.0;
	if (t_m <= 2.05e-225)
		tmp = t_2;
	elseif (t_m <= 5.4e-205)
		tmp = 1.0;
	elseif (t_m <= 2e-171)
		tmp = t_2;
	elseif (t_m <= 1.5e-124)
		tmp = 1.0 + (-1.0 / x);
	elseif (t_m <= 4.1e-97)
		tmp = t_m / ((x ^ -0.5) * l_m);
	else
		tmp = 1.0 + ((0.5 / (x ^ 2.0)) + ((-0.5 / (x ^ 3.0)) + (-1.0 / x)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-225], t$95$2, If[LessEqual[t$95$m, 5.4e-205], 1.0, If[LessEqual[t$95$m, 2e-171], t$95$2, If[LessEqual[t$95$m, 1.5e-124], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4.1e-97], N[(t$95$m / N[(N[Power[x, -0.5], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if t < 2.05000000000000011e-225 or 5.4000000000000002e-205 < t < 2e-171

   1. Initial program 30.1%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in x around inf 59.3%

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

   6. Taylor expanded in l around inf 22.1%

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

      1. *-commutative 22.1%

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

   8. Simplified 22.1%

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

      1. *-un-lft-identity 22.1%

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

      2. pow1/2 22.1%

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

      3. inv-pow 22.1%

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

      4. pow-pow 22.1%

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

      5. metadata-eval 22.1%

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

   10. Applied egg-rr 22.1%

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

       1. *-lft-identity 22.1%

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

   12. Simplified 22.1%

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

       1. associate-*r* 22.1%

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

       2. associate-*r/ 22.1%

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

       3. times-frac 22.1%

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

       4. sqrt-undiv 22.1%

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

       5. metadata-eval 22.1%

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

       6. metadata-eval 22.1%

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

       7. metadata-eval 22.1%

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

       8. times-frac 22.1%

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

       9. *-un-lft-identity 22.1%

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

       10. *-un-lft-identity 22.1%

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

   14. Applied egg-rr 22.1%

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

       1. *-lft-identity 22.1%

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

       2. associate-*r/ 22.1%

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

       3. times-frac 22.1%

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

       4. associate-*l/ 22.1%

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

       5. *-lft-identity 22.1%

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

   16. Simplified 22.1%

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

   if 2.05000000000000011e-225 < t < 5.4000000000000002e-205

   1. Initial program 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in t around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

   if 2e-171 < t < 1.5e-124

   1. Initial program 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in t around inf 59.7%

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

      1. +-commutative 59.7%

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

      2. sub-neg 59.7%

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

      3. metadata-eval 59.7%

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

      4. +-commutative 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in x around inf 59.9%

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

   if 1.5e-124 < t < 4.09999999999999993e-97

   1. Initial program 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in x around inf 80.0%

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

   6. Taylor expanded in l around inf 3.1%

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

      1. *-commutative 3.1%

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

   8. Simplified 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. pow1/2 3.1%

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

      3. inv-pow 3.1%

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

      4. pow-pow 3.1%

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

      5. metadata-eval 3.1%

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

   10. Applied egg-rr 3.1%

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

       1. *-lft-identity 3.1%

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

   12. Simplified 3.1%

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

       1. associate-*r* 3.1%

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

       2. associate-*r/ 3.1%

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

       3. times-frac 3.1%

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

       4. sqrt-undiv 3.1%

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

       5. metadata-eval 3.1%

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

       6. metadata-eval 3.1%

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

       7. metadata-eval 3.1%

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

       8. times-frac 3.1%

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

       9. *-un-lft-identity 3.1%

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

       10. *-un-lft-identity 3.1%

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

   14. Applied egg-rr 3.1%

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

   if 4.09999999999999993e-97 < t

   1. Initial program 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in t around inf 87.4%

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

      1. +-commutative 87.4%

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

      2. sub-neg 87.4%

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

      3. metadata-eval 87.4%

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

      4. +-commutative 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in x around inf 87.6%

      \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 87.6%

         \[\leadsto \left(1 + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      2. metadata-eval 87.6%

         \[\leadsto \left(1 + \frac{\color{blue}{0.5}}{{x}^{2}}\right) - \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right) \]

      3. +-commutative 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \color{blue}{\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]

      4. associate-*r/ 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) \]

      5. metadata-eval 87.6%

         \[\leadsto \left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right) \]

   10. Simplified 87.6%

       \[\leadsto \color{blue}{\left(1 + \frac{0.5}{{x}^{2}}\right) - \left(\frac{1}{x} + \frac{0.5}{{x}^{3}}\right)} \]

   11. Taylor expanded in x around 0 87.6%

       \[\leadsto \color{blue}{\left(1 + 0.5 \cdot \frac{1}{{x}^{2}}\right) - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)} \]
   12. Step-by-step derivation

       1. associate--l+ 87.6%

          \[\leadsto \color{blue}{1 + \left(0.5 \cdot \frac{1}{{x}^{2}} - \left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

       2. sub-neg 87.6%

          \[\leadsto 1 + \color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)} \]

       3. associate-*r/ 87.6%

          \[\leadsto 1 + \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       4. metadata-eval 87.6%

          \[\leadsto 1 + \left(\frac{\color{blue}{0.5}}{{x}^{2}} + \left(-\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right) \]

       5. +-commutative 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(-\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)}\right)\right) \]

       6. distribute-neg-in 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \color{blue}{\left(\left(-0.5 \cdot \frac{1}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)}\right) \]

       7. associate-*r/ 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       8. metadata-eval 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\left(-\frac{\color{blue}{0.5}}{{x}^{3}}\right) + \left(-\frac{1}{x}\right)\right)\right) \]

       9. distribute-neg-frac 87.6%

          \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\color{blue}{\frac{-0.5}{{x}^{3}}} + \left(-\frac{1}{x}\right)\right)\right) \]

       10. metadata-eval 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{\color{blue}{-0.5}}{{x}^{3}} + \left(-\frac{1}{x}\right)\right)\right) \]

       11. distribute-neg-frac 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \color{blue}{\frac{-1}{x}}\right)\right) \]

       12. metadata-eval 87.6%

           \[\leadsto 1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{\color{blue}{-1}}{x}\right)\right) \]

   13. Simplified 87.6%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{t}{{x}^{-0.5} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{0.5}{{x}^{2}} + \left(\frac{-0.5}{{x}^{3}} + \frac{-1}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{{x}^{-0.5} \cdot l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-225}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 7.4 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 3.9 \cdot 10^{-171} \lor \neg \left(t\_m \leq 1.1 \cdot 10^{-121}\right) \land t\_m \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ t_m (* (pow x -0.5) l_m))))
   (*
    t_s
    (if (<= t_m 2.05e-225)
      t_2
      (if (<= t_m 7.4e-205)
        1.0
        (if (or (<= t_m 3.9e-171)
                (and (not (<= t_m 1.1e-121)) (<= t_m 3.5e-98)))
          t_2
          (+ 1.0 (/ -1.0 x))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m / (pow(x, -0.5) * l_m);
	double tmp;
	if (t_m <= 2.05e-225) {
		tmp = t_2;
	} else if (t_m <= 7.4e-205) {
		tmp = 1.0;
	} else if ((t_m <= 3.9e-171) || (!(t_m <= 1.1e-121) && (t_m <= 3.5e-98))) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m / ((x ** (-0.5d0)) * l_m)
    if (t_m <= 2.05d-225) then
        tmp = t_2
    else if (t_m <= 7.4d-205) then
        tmp = 1.0d0
    else if ((t_m <= 3.9d-171) .or. (.not. (t_m <= 1.1d-121)) .and. (t_m <= 3.5d-98)) then
        tmp = t_2
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m / (Math.pow(x, -0.5) * l_m);
	double tmp;
	if (t_m <= 2.05e-225) {
		tmp = t_2;
	} else if (t_m <= 7.4e-205) {
		tmp = 1.0;
	} else if ((t_m <= 3.9e-171) || (!(t_m <= 1.1e-121) && (t_m <= 3.5e-98))) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m / (math.pow(x, -0.5) * l_m)
	tmp = 0
	if t_m <= 2.05e-225:
		tmp = t_2
	elif t_m <= 7.4e-205:
		tmp = 1.0
	elif (t_m <= 3.9e-171) or (not (t_m <= 1.1e-121) and (t_m <= 3.5e-98)):
		tmp = t_2
	else:
		tmp = 1.0 + (-1.0 / x)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m / Float64((x ^ -0.5) * l_m))
	tmp = 0.0
	if (t_m <= 2.05e-225)
		tmp = t_2;
	elseif (t_m <= 7.4e-205)
		tmp = 1.0;
	elseif ((t_m <= 3.9e-171) || (!(t_m <= 1.1e-121) && (t_m <= 3.5e-98)))
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m / ((x ^ -0.5) * l_m);
	tmp = 0.0;
	if (t_m <= 2.05e-225)
		tmp = t_2;
	elseif (t_m <= 7.4e-205)
		tmp = 1.0;
	elseif ((t_m <= 3.9e-171) || (~((t_m <= 1.1e-121)) && (t_m <= 3.5e-98)))
		tmp = t_2;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m / N[(N[Power[x, -0.5], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-225], t$95$2, If[LessEqual[t$95$m, 7.4e-205], 1.0, If[Or[LessEqual[t$95$m, 3.9e-171], And[N[Not[LessEqual[t$95$m, 1.1e-121]], $MachinePrecision], LessEqual[t$95$m, 3.5e-98]]], t$95$2, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 2.05000000000000011e-225 or 7.4000000000000002e-205 < t < 3.8999999999999998e-171 or 1.10000000000000011e-121 < t < 3.5000000000000002e-98

   1. Initial program 30.1%

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

   3. Simplified 30.1%

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

   5. Taylor expanded in x around inf 59.6%

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

   6. Taylor expanded in l around inf 21.8%

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

      1. *-commutative 21.8%

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

   8. Simplified 21.8%

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

      1. *-un-lft-identity 21.8%

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

      2. pow1/2 21.8%

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

      3. inv-pow 21.8%

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

      4. pow-pow 21.8%

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

      5. metadata-eval 21.8%

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

   10. Applied egg-rr 21.8%

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

       1. *-lft-identity 21.8%

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

   12. Simplified 21.8%

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

       1. associate-*r* 21.8%

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

       2. associate-*r/ 21.8%

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

       3. times-frac 21.9%

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

       4. sqrt-undiv 21.9%

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

       5. metadata-eval 21.9%

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

       6. metadata-eval 21.9%

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

       7. metadata-eval 21.9%

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

       8. times-frac 21.9%

          \[\leadsto \color{blue}{\frac{1 \cdot t}{1 \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       9. *-un-lft-identity 21.9%

          \[\leadsto \frac{\color{blue}{t}}{1 \cdot \left(\ell \cdot {x}^{-0.5}\right)} \]

       10. *-un-lft-identity 21.9%

           \[\leadsto \frac{t}{\color{blue}{\ell \cdot {x}^{-0.5}}} \]

   14. Applied egg-rr 21.9%

       \[\leadsto \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}} \]

   if 2.05000000000000011e-225 < t < 7.4000000000000002e-205

   1. Initial program 3.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Simplified 100.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]

   if 3.8999999999999998e-171 < t < 1.10000000000000011e-121 or 3.5000000000000002e-98 < t

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 84.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. +-commutative 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Simplified 84.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Taylor expanded in x around inf 84.3%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-225}:\\ \;\;\;\;\frac{t}{{x}^{-0.5} \cdot \ell}\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{-205}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-171} \lor \neg \left(t \leq 1.1 \cdot 10^{-121}\right) \land t \leq 3.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{t}{{x}^{-0.5} \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{t\_m}{{x}^{-0.5}}}{l\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.05 \cdot 10^{-225}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 3.25 \cdot 10^{-202}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 6.5 \cdot 10^{-122} \lor \neg \left(t\_m \leq 4 \cdot 10^{-97}\right):\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{{x}^{-0.5} \cdot l\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (/ (/ t_m (pow x -0.5)) l_m)))
   (*
    t_s
    (if (<= t_m 2.05e-225)
      t_2
      (if (<= t_m 3.25e-202)
        1.0
        (if (<= t_m 4.1e-171)
          t_2
          (if (or (<= t_m 6.5e-122) (not (<= t_m 4e-97)))
            (+ 1.0 (/ -1.0 x))
            (/ t_m (* (pow x -0.5) l_m)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 2.05e-225) {
		tmp = t_2;
	} else if (t_m <= 3.25e-202) {
		tmp = 1.0;
	} else if (t_m <= 4.1e-171) {
		tmp = t_2;
	} else if ((t_m <= 6.5e-122) || !(t_m <= 4e-97)) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m / (pow(x, -0.5) * l_m);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / (x ** (-0.5d0))) / l_m
    if (t_m <= 2.05d-225) then
        tmp = t_2
    else if (t_m <= 3.25d-202) then
        tmp = 1.0d0
    else if (t_m <= 4.1d-171) then
        tmp = t_2
    else if ((t_m <= 6.5d-122) .or. (.not. (t_m <= 4d-97))) then
        tmp = 1.0d0 + ((-1.0d0) / x)
    else
        tmp = t_m / ((x ** (-0.5d0)) * l_m)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / Math.pow(x, -0.5)) / l_m;
	double tmp;
	if (t_m <= 2.05e-225) {
		tmp = t_2;
	} else if (t_m <= 3.25e-202) {
		tmp = 1.0;
	} else if (t_m <= 4.1e-171) {
		tmp = t_2;
	} else if ((t_m <= 6.5e-122) || !(t_m <= 4e-97)) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = t_m / (Math.pow(x, -0.5) * l_m);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / math.pow(x, -0.5)) / l_m
	tmp = 0
	if t_m <= 2.05e-225:
		tmp = t_2
	elif t_m <= 3.25e-202:
		tmp = 1.0
	elif t_m <= 4.1e-171:
		tmp = t_2
	elif (t_m <= 6.5e-122) or not (t_m <= 4e-97):
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = t_m / (math.pow(x, -0.5) * l_m)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / (x ^ -0.5)) / l_m)
	tmp = 0.0
	if (t_m <= 2.05e-225)
		tmp = t_2;
	elseif (t_m <= 3.25e-202)
		tmp = 1.0;
	elseif (t_m <= 4.1e-171)
		tmp = t_2;
	elseif ((t_m <= 6.5e-122) || !(t_m <= 4e-97))
		tmp = Float64(1.0 + Float64(-1.0 / x));
	else
		tmp = Float64(t_m / Float64((x ^ -0.5) * l_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / (x ^ -0.5)) / l_m;
	tmp = 0.0;
	if (t_m <= 2.05e-225)
		tmp = t_2;
	elseif (t_m <= 3.25e-202)
		tmp = 1.0;
	elseif (t_m <= 4.1e-171)
		tmp = t_2;
	elseif ((t_m <= 6.5e-122) || ~((t_m <= 4e-97)))
		tmp = 1.0 + (-1.0 / x);
	else
		tmp = t_m / ((x ^ -0.5) * l_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 2.05e-225], t$95$2, If[LessEqual[t$95$m, 3.25e-202], 1.0, If[LessEqual[t$95$m, 4.1e-171], t$95$2, If[Or[LessEqual[t$95$m, 6.5e-122], N[Not[LessEqual[t$95$m, 4e-97]], $MachinePrecision]], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$m / N[(N[Power[x, -0.5], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < 2.05000000000000011e-225 or 3.24999999999999978e-202 < t < 4.1e-171

   1. Initial program 30.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 30.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 59.3%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

   6. Taylor expanded in l around inf 22.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}}} \]
   7. Step-by-step derivation

      1. *-commutative 22.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{\frac{1}{x}}} \]

   8. Simplified 22.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 22.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{x}}\right)}} \]

      2. pow1/2 22.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.5}}\right)} \]

      3. inv-pow 22.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot {\color{blue}{\left({x}^{-1}\right)}}^{0.5}\right)} \]

      4. pow-pow 22.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot \color{blue}{{x}^{\left(-1 \cdot 0.5\right)}}\right)} \]

      5. metadata-eval 22.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot {x}^{\color{blue}{-0.5}}\right)} \]

   10. Applied egg-rr 22.1%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)}} \]
   11. Step-by-step derivation

       1. *-lft-identity 22.1%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{{x}^{-0.5}}} \]

   12. Simplified 22.1%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{{x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. associate-*r* 22.1%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{2} \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       2. associate-*r/ 22.1%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       3. times-frac 22.1%

          \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

       4. sqrt-undiv 22.1%

          \[\leadsto \color{blue}{\sqrt{\frac{2}{2}}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       5. metadata-eval 22.1%

          \[\leadsto \sqrt{\color{blue}{1}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       6. metadata-eval 22.1%

          \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       7. metadata-eval 22.1%

          \[\leadsto \color{blue}{\frac{1}{1}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       8. times-frac 22.1%

          \[\leadsto \color{blue}{\frac{1 \cdot t}{1 \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       9. *-un-lft-identity 22.1%

          \[\leadsto \frac{\color{blue}{t}}{1 \cdot \left(\ell \cdot {x}^{-0.5}\right)} \]

       10. *-un-lft-identity 22.1%

           \[\leadsto \frac{t}{\color{blue}{\ell \cdot {x}^{-0.5}}} \]

   14. Applied egg-rr 22.1%

       \[\leadsto \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}} \]
   15. Step-by-step derivation

       1. *-lft-identity 22.1%

          \[\leadsto \color{blue}{1 \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

       2. associate-*r/ 22.1%

          \[\leadsto \color{blue}{\frac{1 \cdot t}{\ell \cdot {x}^{-0.5}}} \]

       3. times-frac 22.1%

          \[\leadsto \color{blue}{\frac{1}{\ell} \cdot \frac{t}{{x}^{-0.5}}} \]

       4. associate-*l/ 22.1%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{t}{{x}^{-0.5}}}{\ell}} \]

       5. *-lft-identity 22.1%

          \[\leadsto \frac{\color{blue}{\frac{t}{{x}^{-0.5}}}}{\ell} \]

   16. Simplified 22.1%

       \[\leadsto \color{blue}{\frac{\frac{t}{{x}^{-0.5}}}{\ell}} \]

   if 2.05000000000000011e-225 < t < 3.24999999999999978e-202

   1. Initial program 3.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Simplified 100.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]

   if 4.1e-171 < t < 6.49999999999999965e-122 or 4.00000000000000014e-97 < t

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 84.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. +-commutative 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Simplified 84.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Taylor expanded in x around inf 84.3%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]

   if 6.49999999999999965e-122 < t < 4.00000000000000014e-97

   1. Initial program 32.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 32.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 80.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

   6. Taylor expanded in l around inf 3.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}}} \]
   7. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{\frac{1}{x}}} \]

   8. Simplified 3.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 3.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\left(1 \cdot \sqrt{\frac{1}{x}}\right)}} \]

      2. pow1/2 3.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.5}}\right)} \]

      3. inv-pow 3.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot {\color{blue}{\left({x}^{-1}\right)}}^{0.5}\right)} \]

      4. pow-pow 3.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot \color{blue}{{x}^{\left(-1 \cdot 0.5\right)}}\right)} \]

      5. metadata-eval 3.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \left(1 \cdot {x}^{\color{blue}{-0.5}}\right)} \]

   10. Applied egg-rr 3.1%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{\left(1 \cdot {x}^{-0.5}\right)}} \]
   11. Step-by-step derivation

       1. *-lft-identity 3.1%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{{x}^{-0.5}}} \]

   12. Simplified 3.1%

       \[\leadsto \sqrt{2} \cdot \frac{t}{\left(\sqrt{2} \cdot \ell\right) \cdot \color{blue}{{x}^{-0.5}}} \]
   13. Step-by-step derivation

       1. associate-*r* 3.1%

          \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\sqrt{2} \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       2. associate-*r/ 3.1%

          \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot t}{\sqrt{2} \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       3. times-frac 3.1%

          \[\leadsto \color{blue}{\frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}}} \]

       4. sqrt-undiv 3.1%

          \[\leadsto \color{blue}{\sqrt{\frac{2}{2}}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       5. metadata-eval 3.1%

          \[\leadsto \sqrt{\color{blue}{1}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       6. metadata-eval 3.1%

          \[\leadsto \color{blue}{1} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       7. metadata-eval 3.1%

          \[\leadsto \color{blue}{\frac{1}{1}} \cdot \frac{t}{\ell \cdot {x}^{-0.5}} \]

       8. times-frac 3.1%

          \[\leadsto \color{blue}{\frac{1 \cdot t}{1 \cdot \left(\ell \cdot {x}^{-0.5}\right)}} \]

       9. *-un-lft-identity 3.1%

          \[\leadsto \frac{\color{blue}{t}}{1 \cdot \left(\ell \cdot {x}^{-0.5}\right)} \]

       10. *-un-lft-identity 3.1%

           \[\leadsto \frac{t}{\color{blue}{\ell \cdot {x}^{-0.5}}} \]

   14. Applied egg-rr 3.1%

       \[\leadsto \color{blue}{\frac{t}{\ell \cdot {x}^{-0.5}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.05 \cdot 10^{-225}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-202}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{t}{{x}^{-0.5}}}{\ell}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-122} \lor \neg \left(t \leq 4 \cdot 10^{-97}\right):\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{{x}^{-0.5} \cdot \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{l\_m} \cdot \sqrt{x}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-225}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_m \leq 4.7 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{-172} \lor \neg \left(t\_m \leq 1.06 \cdot 10^{-121}\right) \land t\_m \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m l_m) (sqrt x))))
   (*
    t_s
    (if (<= t_m 1.55e-225)
      t_2
      (if (<= t_m 4.7e-197)
        1.0
        (if (or (<= t_m 6e-172)
                (and (not (<= t_m 1.06e-121)) (<= t_m 2.4e-98)))
          t_2
          (+ 1.0 (/ -1.0 x))))))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / l_m) * sqrt(x);
	double tmp;
	if (t_m <= 1.55e-225) {
		tmp = t_2;
	} else if (t_m <= 4.7e-197) {
		tmp = 1.0;
	} else if ((t_m <= 6e-172) || (!(t_m <= 1.06e-121) && (t_m <= 2.4e-98))) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (t_m / l_m) * sqrt(x)
    if (t_m <= 1.55d-225) then
        tmp = t_2
    else if (t_m <= 4.7d-197) then
        tmp = 1.0d0
    else if ((t_m <= 6d-172) .or. (.not. (t_m <= 1.06d-121)) .and. (t_m <= 2.4d-98)) then
        tmp = t_2
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = (t_m / l_m) * Math.sqrt(x);
	double tmp;
	if (t_m <= 1.55e-225) {
		tmp = t_2;
	} else if (t_m <= 4.7e-197) {
		tmp = 1.0;
	} else if ((t_m <= 6e-172) || (!(t_m <= 1.06e-121) && (t_m <= 2.4e-98))) {
		tmp = t_2;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = (t_m / l_m) * math.sqrt(x)
	tmp = 0
	if t_m <= 1.55e-225:
		tmp = t_2
	elif t_m <= 4.7e-197:
		tmp = 1.0
	elif (t_m <= 6e-172) or (not (t_m <= 1.06e-121) and (t_m <= 2.4e-98)):
		tmp = t_2
	else:
		tmp = 1.0 + (-1.0 / x)
	return t_s * tmp

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(Float64(t_m / l_m) * sqrt(x))
	tmp = 0.0
	if (t_m <= 1.55e-225)
		tmp = t_2;
	elseif (t_m <= 4.7e-197)
		tmp = 1.0;
	elseif ((t_m <= 6e-172) || (!(t_m <= 1.06e-121) && (t_m <= 2.4e-98)))
		tmp = t_2;
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = (t_m / l_m) * sqrt(x);
	tmp = 0.0;
	if (t_m <= 1.55e-225)
		tmp = t_2;
	elseif (t_m <= 4.7e-197)
		tmp = 1.0;
	elseif ((t_m <= 6e-172) || (~((t_m <= 1.06e-121)) && (t_m <= 2.4e-98)))
		tmp = t_2;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.55e-225], t$95$2, If[LessEqual[t$95$m, 4.7e-197], 1.0, If[Or[LessEqual[t$95$m, 6e-172], And[N[Not[LessEqual[t$95$m, 1.06e-121]], $MachinePrecision], LessEqual[t$95$m, 2.4e-98]]], t$95$2, N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.54999999999999998e-225 or 4.7000000000000001e-197 < t < 5.99999999999999967e-172 or 1.05999999999999997e-121 < t < 2.40000000000000005e-98

   1. Initial program 30.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 30.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 59.6%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\sqrt{\color{blue}{\left(2 \cdot \frac{{t}^{2}}{x} + \left(2 \cdot {t}^{2} + \frac{{\ell}^{2}}{x}\right)\right) - -1 \cdot \frac{2 \cdot {t}^{2} + {\ell}^{2}}{x}}}} \]

   6. Taylor expanded in l around inf 21.8%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\ell \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1}{x}}}} \]
   7. Step-by-step derivation

      1. *-commutative 21.8%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \ell\right)} \cdot \sqrt{\frac{1}{x}}} \]

   8. Simplified 21.8%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(\sqrt{2} \cdot \ell\right) \cdot \sqrt{\frac{1}{x}}}} \]

   9. Taylor expanded in t around 0 18.9%

      \[\leadsto \color{blue}{\frac{t}{\ell} \cdot \sqrt{x}} \]

   if 1.54999999999999998e-225 < t < 4.7000000000000001e-197

   1. Initial program 3.1%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 3.1%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 100.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 100.0%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Simplified 100.0%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{1} \]

   if 5.99999999999999967e-172 < t < 1.05999999999999997e-121 or 2.40000000000000005e-98 < t

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

   3. Simplified 39.5%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 84.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
   6. Step-by-step derivation

      1. +-commutative 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

      2. sub-neg 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

      3. metadata-eval 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

      4. +-commutative 84.1%

         \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

   7. Simplified 84.1%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

   8. Taylor expanded in x around inf 84.3%

      \[\leadsto \color{blue}{1 - \frac{1}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-225}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-172} \lor \neg \left(t \leq 1.06 \cdot 10^{-121}\right) \land t \leq 2.4 \cdot 10^{-98}:\\ \;\;\;\;\frac{t}{\ell} \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.3% accurate, 45.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

3. Simplified 33.9%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 39.9%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
6. Step-by-step derivation

   1. +-commutative 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

   2. sub-neg 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

   3. metadata-eval 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   4. +-commutative 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

7. Simplified 39.9%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

8. Taylor expanded in x around inf 39.9%

   \[\leadsto \color{blue}{1 - \frac{1}{x}} \]

9. Final simplification 39.9%

   \[\leadsto 1 + \frac{-1}{x} \]
10. Add Preprocessing

Alternative 14: 75.6% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 33.9%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]

3. Simplified 33.9%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \frac{t}{\sqrt{\frac{x + 1}{x + -1} \cdot \mathsf{fma}\left(2, t \cdot t, \ell \cdot \ell\right) - \ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 39.9%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{1 + x}{x - 1}}}} \]
6. Step-by-step derivation

   1. +-commutative 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{\color{blue}{x + 1}}{x - 1}}} \]

   2. sub-neg 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{x + \left(-1\right)}}}} \]

   3. metadata-eval 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{x + \color{blue}{-1}}}} \]

   4. +-commutative 39.9%

      \[\leadsto \sqrt{2} \cdot \frac{t}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{\color{blue}{-1 + x}}}} \]

7. Simplified 39.9%

   \[\leadsto \sqrt{2} \cdot \frac{t}{\color{blue}{\left(t \cdot \sqrt{2}\right) \cdot \sqrt{\frac{x + 1}{-1 + x}}}} \]

8. Taylor expanded in x around inf 39.5%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 39.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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)))))